Papers from 2025

This is the second of two papers studying both the geometric structure of Fano fibrations and the application to Kähler-Ricci flows developing a singularity in finite time. We assume that the Kähler-Ricci flow on a compact Kähler manifold has a rational initial metric and develops a singularity in finite time such that the manifold admits a Fano fibration structure. Moreover, it is assumed that the volume form of the flow collapses uniformly at the rate of $C^{-1}(T-t{)}^{n-m}\Omega \le \omega (t{)}^n\le C(T-t{)}^{n-m}\Omega$. Under this setting, a diameter bound is obtained in any compact set away from singular fibres and the diameter of the fibres is proven to collapse at the optimal rate $\sqrt{T-t}$. Furthermore, several precise $C^0$-estimates are proven for the potential of the complex Monge-Ampere flow which involve the potentials of singular twisted Kähler-Einstein metrics on the base variety from part I. Finally, in the case of Kähler-Einstein Fano fibres, we deduce Type I scalar curvature in any compact set away from singular fibres and globally for a submersion.

This is the first of two papers studying both the geometric structure of Fano fibrations and the application to Kähler-Ricci flows developing a singularity in finite time. Given a Fano fibration which is generated by Kawamata's theorem from a compact Kähler manifold $X$ endowed with an ample, rational line bundle $L$ and non-nef canonical line bundle $K_X$, we construct a $(1,1)$-form on the regular part of the base analytic variety which is related to the Weil-Petersson metric. It is also proven that the singular Kähler metric constructed by Zhang, Zhang, on the base analytic variety satisfies a twisted Kähler-Einstein equation involving this $(1,1)$-form and, for a submersion, that the Chern classes of $X$ and the base manifold decompose in terms of this $(1,1)$-form.

We study a spinor-driven formulation of geometric evolution on closed $3$-manifolds, in which the spinor field is treated as the primary dynamical variable and the Riemannian metric is induced conformally by the spinor amplitude. We introduce a spinorial heat flow governed by the squared Dirac operator, $${\partial }_t\psi =-D_{g(\psi )}^2\psi ,$$ where the metric $g(\psi )$ depends nonlinearly on the evolving spinor field. As a consequence, the resulting system is quasi-linear and parabolic away from the nodal set $\{\psi =0\}$, while exhibiting degenerate behavior at vanishing spinor amplitude. We show that degeneration of the induced metric corresponds analytically to nodal behavior of the spinor field, rather than to curvature blow-up of the spinor evolution itself. This observation motivates an interpretation of geometric singularities as spinorial nodal transitions, across which the spinor field remains locally bounded in a weak or weighted sense. The induced metric evolution is derived explicitly and shown to be purely conformal, capturing only the trace component of curvature evolution and containing additional gradient terms that are not controlled a priori. Accordingly, the proposed flow should not be identified with the Ricci flow, and any analogy with curvature smoothing is understood at a heuristic level. The present work establishes a coherent analytical framework for studying geometric degeneration via spinor dynamics and highlights several open problems in degenerate parabolic theory, including rigorous existence results and the precise role of nodal structures in geometric and topological evolution.

We show that any toric asymptotically conical shrinking gradient Kähler-Ricci soliton on an anti-canonically polarised resolution of a Kähler cone satisfies a complex Monge-Ampère equation. We then set up an Aubin continuity path to solve the resulting equation and show that it has a solution at the initial value of the path parameter in the toric case. This we do by implementing another continuity method. Finally, we prove openness of the initial value of the path parameter independent of the toricity.

In this paper, we revisit the study of almost Ricci-Bourguignon solitons by clarifying their position in the broader context of Einstein-type metrics. Motivated by known rigidity results for compact almost Ricci solitons, we aim to identify conditions under which a compact almost RB-soliton is trivial or exhibits special geometric properties. We compare our results with classical theorems of Barros and Ribeiro, and explain explicitly how our work extends or complements these earlier findings.

We study model semilinear equations on complete and non-compact weighted Riemannian manifolds with non-negative Bakry-Émery Ricci curvature. Our main goal is to classify positive solutions of the equation at the Sobolev-critical exponent, and furthermore to prove that the existence of such solutions implies rigidity of the manifold and triviality of the weight. This is possible when the weighted manifold has non-negative finite dimensional Bakry-Émery Ricci curvature, and even under the weaker condition of non-negative infinite dimensional Bakry-Émery Ricci curvature, up to imposing some additional conditions in the latter case. To exhibit the sharpness of these additional conditions, we construct a non-trivial positive solution of the critical problem on a weighted manifold with positive infinite dimensional curvature. We also obtain a corresponding rigidity result for solutions of the Liouville equation on weighted Riemannian surfaces. Finally, we prove some non-existence theorems when the nonlinearity is sub-critical or simply under certain volume growth conditions. In particular, the latter rules out all positive solutions on shrinking gradient Ricci solitons.

We establish global existence of the pluriclosed flow with arbitrary initial data on Oeljeklaus-Toma manifolds, and Gromov-Hausdorff convergence of blowdown limits to a torus under natural conjectural bounds on the flow at infinity. In the case of generalized Kähler-Ricci flow we prove refined a priori estimates in support of these conjectural bounds.

In this paper, we demonstrate certain curvature estimates on complete non-compact steady and expanding gradient Ricci solitons in higher dimensions. In the expanding case, we prove that if the Ricci curvature decays at least quadratically, then the curvature operator decays at the rate $\text{BigO}(1/r^2)$ when $n=4$ and $\text{BigO}((\log r)/r^2)$ when $n\ge 5$. This refines the curvature bounds in a previous result by Cao-Liu-Xie, and removes the nonnegative Ricci curvature assumption in the estimates by Cao-Liu and Cao-Liu-Xie. As a geometric application, we establish the existence and uniqueness of $C^{1,\alpha }$ conical structure at infinity of Ricci expander with finite Ricci curvature ratio. In the steady case, using an integral estimate of the curvature, we prove that the curvature operator has at most polynomial growth when the potential function is proper and the Ricci curvature has linear decay. Moreover, we also confirm that the curvature is bounded if we further assume the Ricci curvature has super-linear decay $\text{BigO}(r^{-1-\epsilon })$. As an application, we prove the existence and uniqueness of cylindrical structure at infinity of steady soliton with super-linear Ricci curvature decay and proper potential function.

This paper introduce the idea of second Ricci solitons. A second Ricci soliton is nothing but a steady hyperbolic Ricci soliton. We study the geometry of closed and compact second Ricci soliton manifolds. Immersed submanifolds as second solitons also will be investigated. Finally, we investigate this structure on warped product manifolds.

In this paper, we study Perelman' s $\mathcal{W}$ entropy for mean curvature flow in ${\mathbb{R}}^{n+1}$. Analogously to Perelman's $\mathcal{W}$-entropy defined for Ricci flow, K. Ecker in defined a functional $\mathcal{W}$ for the mean curvature flow in ${\mathbb{R}}^{n+1}$ and the region it encloses, and made the conjecture that this functional is monotonically increasing in time. We modify K. Ecker's definition and, using Hamilton's Harnack inequality for mean curvature flow, prove that our redefined $\mathcal{W}$-entropy is monotonically decreasing in time. Additionally, we provide a rigidity theorem for this $\mathcal{W}$-entropy.

We study flows of $G_2$-structures guided by the principle of dimensional reduction: natural geometric flows in $G_2$-geometry reduce to natural flows in complex geometry. Our main examples are the $G_2$-Laplacian coflow, which lifts the Kähler–Ricci flow, and a 7-dimensional lift of the anomaly flow on complex threefolds. The $G_2$-lift of the anomaly flow deforms conformally coclosed $G_2$-structures. We compare the $G_2$-anomaly flow to the $G_2$-Laplacian coflow, and investigate short-time existence and fixed points.

String backgrounds, defined here as metric connections with skew-symmetric torsion and reduced holonomy, yield generalized Ricci solitons relative to the Lee vector field. By a variational argument using the string action, they are also gradient generalized Ricci solitons relative to a potential function. These two observations combine to yield a canonical symmetry, and in this work we derive fundamental features of the transverse geometry, and rigidity phenomena. We prove in a unified conceptual fashion that the transverse geometry satisfies the string generalized Ricci soliton equations (a simplified Hull-Strominger system) in many settings including almost Hermitian, almost contact, $SU(3)$, $G_2$, and $\mathrm{Spin}(7)$ geometry. We also show that the transverse geometry is always conformally co-closed, with the conformal factor given by the associated soliton potential.

We show that the recently discovered BCCD shrinking soliton is linearly unstable, by extending the approach of and, via recent work the on gradient shrinking Ricci solitons. On the other hand, we prove that the weighted $L^2$-spectra of the weighted Lichnerowicz Laplacians of steady and expanding Kähler Ricci solitons are nonpositive in real dimension $4$. We additionally determine the linear stability of the orbifold singularities of Kähler solitons: shrinkers are unstable, steadies are neutrally stable and expanders are strictly stable. All of these results follow from new Weitzenböck formulae for the weighted Lichnerowicz Laplacian specialized to Kähler metrics.

We prove a weaker version of the transcendental base-point freeness on compact Kähler manifolds. As a consequence, we derive the diameter lower bound for finite time singularities of Kähler-Ricci flow with non-Fano initial data.

Given a compact Kähler manifold $X$ and a closed, positive $(1,1)$-current $T$ on $X$, we find sufficient conditions for $T$ to induce a metric structure $(X,d_T)$ which is the Gromov-Hausdorff limit of compact Kähler manifolds either in a "static" way or at time zero of smooth Kähler-Ricci flows. In dimension $1$ we extend works of T. Richard and M. Simon, showing that any oriented compact Alexandrov surface with bounded integral curvature and without cusp is the initial datum of a Kähler-Ricci flow.

Quasi-Einstein manifolds are well-studied generalizations of Einstein manifolds. This includes gradient Ricci solitons and has a natural correspondence with the warped product Einstein manifolds. A quasi-Einstein metric is said to be rigid when it reduces to an Einstein metric. On a different note, Einstein metrics can be viewed as fixed points of the Ricci flow up to homothety. While gradient Ricci solitons are generalized fixed points of the Ricci flow, not much is known, in general, about the evolution of quasi-Einstein metrics under the Ricci flow. In this paper, we employ an identity associated to the evolution of curvature along the Ricci flow, to conclude the rigidity of certain closed quasi-Einstein manifolds.

We review different notions of synthetic Ricci flow that apply to time-dependent families of metric measure spaces and which are based on properties of the heat flow, ideas from optimal transport, and the asymptotic behaviour of volumes. Each notion equivalently characterises (weighted) Ricci flow for smooth families of weighted Riemannian manifolds. We discuss the features of the different notions on various examples.

We introduce a dynamical energy functional on compact ancient asymptotically Ricci-flat Ricci flows with modest decay using limits of conjugate heat flows. This functional satisfies a steady Ricci breather-type rigidity and provides an upper bound for the ordinary $\lambda$-functional while retaining many of its properties. In addition, motivated by work of Colding and Minicozzi, we derive local eigenvalue estimates for normalized Ricci flows coupled with conjugate heat flows.

In the framework of the study of homogeneous Lorentzian three-manifolds, we consider here the only class of examples which admit a four-dimensional group of isometries but are neither Lorentzian Bianchi-Cartan-Vranceanu spaces nor plane waves. We obtain an explicit description in global coordinates of these special homogeneous Lorentzian manifolds. We then prove that all such examples are non-gradient expanding Ricci solitons.

In 2004, Manning showed that the topological entropy of the geodesic flow of a closed surface of non-constant negative curvature is strictly decreasing along the normalized Ricci flow, and he asked if an analogous result holds in higher dimensions for metrics in a neighborhood of a hyperbolic metric. In this paper, we affirmatively answer this question. Namely, we show that the topological entropy of the geodesic flow of a closed Riemannian manifold that carries a hyperbolic metric is indeed strictly decreasing along the normalized Ricci flow starting from a metric of variable negative sectional curvature sufficiently close to the hyperbolic metric.

The classical Bach tensor in four dimensions can be expressed as a linear combination of two independent, symmetric, divergence-free, quadratic-in-curvature tensors U and V. Several classification results for gradient-shrinking Ricci solitons have been obtained under the assumption that the Bach tensor vanishes. We define a Bach-like tensor to be any other linear combination of U and V. We prove that within a certain cone of parameters, the vanishing of a Bach-like tensor forces a four-dimensional complete gradient-shrinking Ricci soliton to be either Einstein or isometric to the Gaussian soliton, extending the results of Cao–Chen (2013). The special case where U=0 forces $f_{\min }\in \{0,1,2\}$, with rigidity holding when $f_{\min }=0,2$. The remaining case $f_{\min }=1$ is the central open problem, with a cylinder as the conjectured exceptional geometry. Finally, we show that Bach-like tensors arise as Euler–Lagrange equations of a two-parameter family of quadratic curvature functionals and compute the corresponding first and second variation formulas.

We investigate stationary torsional configurations supported by chiral Majorana neutrino currents in linearized gravity. A Ricci-flow-inspired geometric relaxation (with no physical time interpretation) is introduced to drive the metric perturbation toward fixed points sustained by chiral sources while keeping curvature invariants negligible. We show that divergence-free chiral currents can support globally non-trivial torsional holonomy stabilized by topological invariants associated with the fundamental groups pi1(S1) and pi3(S3). Toroidal skyrmionic domains emerge when one chirality dominates, whereas a chiral-flip interference sector enables Moebius-type non-orientable bridges between opposite-chirality regions. In the static limit, a Green-function formulation provides a finite-range Yukawa-type response governed by the neutrino coherence length. These results identify a purely torsional mechanism, independent of local curvature, through which coherent chiral currents may influence effective gravitational behavior in neutrino-rich environments.

We consider noncompact complete Kähler manifolds with nonnegative bisectional curvature. Our main results are: 1. Precise relations among refined minimal degree of polynomial growth holomorphic functions and holomorphic volume forms, $\mathrm{AVR}$ (asymptotic volume ratio) and $\mathrm{ASCD}$ (average of scalar curvature decay) are established. 2. The Lyapunov asymptotic behavior of the Kähler-Ricci flow can be described in terms of polynomial growth holomorphic functions. This provides a unifying perspective that bridges the two distinct proofs of Yau's uniformization conjecture by Liu and Chau-Lee-Tam. These resolve two conjectures made by Yang.

In this paper, we study the singular set $\mathcal{S}$ of a noncollapsed Ricci flow limit space, arising as the pointed Gromov–Hausdorff limit of a sequence of closed Ricci flows with uniformly bounded entropy. The singular set $\mathcal{S}$ admits a natural stratification: \beginequation* \mathcal S^0 \subset \mathcal S^1 \subset \cdots \subset \mathcal S^n-2=\mathcal S, \endequation* where a point $z\in {\mathcal{S}}^k$ if and only if no tangent flow at $z$ is $(k+1)$-symmetric. In general, the Hausdorff dimension of ${\mathcal{S}}^k$ with respect to the spacetime distance is at most $k$. We show that the subset ${\mathcal{S}}_{\mathrm{qc}}^k\subset {\mathcal{S}}^k$, consisting of points where some tangent flow is given by a standard cylinder or its quotient, is parabolic $k$-rectifiable. In dimension four, we prove the stronger statement that each stratum ${\mathcal{S}}^k$ is parabolic $k$-rectifiable for $k\in \{0,1,2\}$. Furthermore, we establish a sharp uniform ${\mathcal{H}}^2$-volume bound for $\mathcal{S}$ and show that, up to a set of ${\mathcal{H}}^2$-measure zero, the tangent flow at any point in $\mathcal{S}$ is backward unique. In addition, we derive $L^1$-curvature bounds for four-dimensional closed Ricci flows. As an application, we resolve Perelman's bounded diameter conjecture for three-dimensional closed Ricci flows.

We prove well-posedness of the analytic Cauchy problem for gradient generalized Ricci solitons on an abelian bundle gerbe and solve the initial data equations on every compact Riemann surface. Along the way, we provide a novel characterization of the self-similar solutions of the generalized Ricci flow by means of families of automorphisms of the underlying abelian bundle gerbe covering families of diffeomorphisms isotopic to the identity.

We review recent results relating linear stability to dynamical stability and the scalar curvature rigidity of Einstein manifolds. We discuss closed and open Einstein manifolds as well as complete noncompact Einstein manifolds which are asymptotically locally Euclidean and asymptotically hyperbolic. For these classes, the relation to the positive mass theorem will also be explained.

The main objective of this article is to study the mean curvature flow into an ambient compact smooth manifold M with boundary and with a Riemannian metric that evolves by a self-similar solution of the Ricci flow coupled with the harmonic map heat flow of a map from M to a Riemannian manifold N. In this context, we address a functional associated with this flow and calculate its variation along parameters that preserve the weighted volume measure. An extension of Hamilton's differential Harnack expression appears by considering the boundary of M evolving by mean curvature flow, which must vanish on the gradient steady soliton case. Next, we obtain a Huisken monotonicity-type formula for the mean curvature flow in the proposed background. We also show how to construct a family of mean curvature solitons and establish a characterization of such a family.

For any $n\ge 4$, we construct an $(n-2)$-parameter family of steady gradient Ricci solitons with non-negative curvature operator and prescribed by the eigenvalues of Ricci tensor at a critical point of the soliton potential. Among them lies an $(n-3)$-parameter subfamily of non-collapsed solitons. These solitons generalized the flying wings constructed by the second named author and produced new examples of steady gradient Ricci solitons with non-negative curvature operator for $n\ge 4$. Our approach is based on constructing continuous families of Ricci flows smoothing emanating from continuous families of spherical polyhedra which still preserves symmetry. This is built upon a new stability result of Ricci flows with scaling invariant estimates. As another application of the method, we prove the stability of asymptotically conical expanding solitons constructed by Deruelle under $L^{\infty }$ perturbation of links. In particular, the $C^0$-convergence of smooth links implies the smooth convergence of the expanding solitons.

We show $L^1$-bounds of the Riemann curvature tensor on a smooth closed $n$-dimensional Ricci flow. To achieve this we introduce the notion of a neck of maximal symmetry, similar to the one in Cheeger-Jiang-Naber and Jiang-Naber and establish a decomposition result by balls with uniform curvature bounds that satisfy an appropriate $(n-2)$-content estimate.

We prove that on ALF $n$-manifolds with $n\ge 4$ the Ricci flow preserves the ALF structure, and develop a weighted Fredholm framework adapted to ALF manifolds. Motivated by Perelman's $\lambda$-functional, we define a renormalized functional ${\lambda }_{\mathrm{ALF}}$ whose gradient flow is the Ricci flow. It is built from a relative mass with respect to a reference Ricci-flat metric at infinity. This yields a natural notion of variational and linear stability for Ricci-flat ALF $4$-metrics and lets us show that the conformally Kähler, non-hyperkähler examples are dynamically unstable along Ricci flow. We finally relate the sign of ${\lambda }_{\mathrm{ALF}}$ to positive relative mass statements for ALF metrics.

In this paper, we establish a Lojasiewicz inequality for the pointed $\mathcal{W}$-entropy in the Ricci flow, under the assumption that the geometry near the base point is close to a standard cylinder ${\mathbb{R}}^k\times S^{n-k}$ or the quotient thereof. As an application, we prove the strong uniqueness of the cylindrical tangent flow at the first singular time of the Ricci flow. Specifically, we show that the modified Ricci flow near the singularity converges to the cylindrical model under a fixed gauge.

We consider the space of smooth gradient expanding Ricci soliton structures on $S^1\times {\mathbb{R}}^3$ and $S^2\times {\mathbb{R}}^2$ which are invariant under the action of $\text{SO}(3)\times \text{SO}(2)$. In the case of each topology, there exists a $2$-parameter family of cohomogeneity one solitons asymptotic to cones over the link $S^2\times S^1$, as constructed by Nienhaus-Wink and Buzano-Dancer-Gallaugher-Wang. By analyzing the resultant soliton ODEs, we reconstruct the $2$-parameter families in each case and provide an alternate proof of conicality. Analogous to work of Bamler and Chen, we define a notion of expander degree for these cohomogeneity one solitons through a properness result. We then proceed to calculate this cohomogeneity one expander degree in the cases of the specific topologies.

The aim of this paper is to study geometrical aspects of static spacetime admitting an almost gradient Ricci soliton. Among others, We first determine the conditions under which the base manifold of static spacetime possess an almost gradient Ricci soliton and we show that the almost gradient Ricci soliton become steady gradient Ricci soliton when static spacetime turns to a vacuum static spacetime. Next, we exhibit that an expanding almost gradient Ricci soliton on base manifold of non-compact and connected static spacetime satisfies shr$\ddot{o}$dinger's equation for a smooth function $f$. Also, we find the soliton constant under which the static perfect fluid spacetime with almost gradient Ricci soliton holds the null convergence condition and the strong energy condition. Further, we study the almost gradient Ricci soliton on base manifold of static perfect fluid spacetime with potential function as warping function and it is shown that the base manifold of a static perfect fluid spacetime with an almost gradient Ricci soliton is an Einstein manifold. Next, we obtain a necessary and sufficient condition on soliton constant to obey timelike convergence condition. Further, we obtain some results for Ricci symmetric and weakly Ricci symmetric base manifold of static perfect fluid spacetime admitting gradient Ricci soliton. Finally, we find the nature of almost gradient Ricci soliton on $4$-dimensional half conformally flat base manifold of static perfect fluid spacetime.

We prove that a three dimensional compact Ricci flow that encounters a Type I singularity has uniformly bounded diameter up to the singular time, thus giving an affirmative answer - for Type I singularities - to a conjecture of Perelman. To achieve this, we introduce a concept of a neck-region for a Ricci flow, analogous to the neck-regions introduced by Jiang-Naber and Cheeger-Jiang-Naber, in the study of Ricci limit spaces. We then prove that the associated packing measure is, in a certain sense, Ahlfors regular, a result that holds in any dimension.

We present a differential geometric formulation of the Poincare problem using the calculus of moving surfaces (CMS). In this framework, an n dimensional compact hypersurface evolves under a velocity field that couples motion to the extrinsic curvature tensor while preserving topology through smooth diffeomorphic flow. A variational energy principle identifies constant mean curvature (CMC) manifolds as the unique stationary equilibria of CMS dynamics. Consequently, the evolution of any compact simply connected hypersurface relaxes to a CMC equilibrium and, in the isotropic case, to the round sphere. Unlike Ricci flow approaches, which are dimension restricted and require topological surgery, the CMS formulation holds for all dimensions and preserves manifold topology for all time. This provides a deterministic geometric mechanical route to the Poincare conclusion, unifying dynamics, topology, and equilibrium geometry within a single analytic framework.

In this article, we investigate a gradient almost Ricci soliton with harmonic Weyl tensor. We first prove that its Ricci tensor has at most three distinct eigenvalues of constant multiplicities in a neighborhood of a regular point of the potential function. Then, we classify those with exactly two distinct eigenvalues. It is worth mentioning that the case with exactly one eigenvalue has already been settled elsewhere. Our results are based on a local representation of these manifolds as multiply warped products of a one-dimensional base, having at most two Einstein fibers, which we also obtain in this paper. These results extend a result by Catino, who assumes, in addition, that the Weyl tensor is radially flat, and a result by Kim, who considers the four-dimensional case.

The objective of this paper is to deepen the study of vector fields on hyperbolic spaces ${\mathbb{H}}^n$ that transform them into a Ricci-Bourguignon soliton. Starting from a recent work in which characterizes these fields as Killing fields of a specific shape, we propose a detailed geometric study of their structure and behavior in the dimensions $n=2,3$ and $n\ge 3$. In odd dimension we show that the dual form of these vectors are contact forms. This work aims to enrich the understanding of self-similar solutions of the Ricci flow in a more general context.

Catino, Mastrolia, Monticelli, and Rigoli have launched an ambitious program to study known geometric solitons from a unified perspective, which they term Einstein-type manifolds. This framework allows one to treat Ricci solitons, Yamabe solitons, and all of their generalizations simultaneously. Einstein-type manifolds are characterized by four constants $\alpha ,\beta ,\mu$ and $\rho$. In this paper, we show that when $\alpha =0$, complete gradient Einstein-type Sasakian manifolds are trivial or isometric to the unit sphere. As a consequence, many geometric solitons on Sasakian manifolds turn out to be trivial or isometric to the unit sphere.

We establish a weak compactness theorem for the moduli space of closed Ricci flows, each equipped with a natural spacetime distance, under pointed Gromov–Hausdorff convergence. For the subspace of flows with uniformly bounded entropy, we further develop a structure theory for the corresponding noncollapsed Ricci flow limit spaces, showing that the regular part, where convergence is smooth, admits the structure of a Ricci flow spacetime, while the singular set has codimension at least four.

In this article, we give a new proof of a result due to J. Kim, which states that the Ricci tensor of a gradient Ricci soliton with dimension $n\ge 4$ and harmonic Weyl tensor has at most three distinct eigenvalues. This result constitutes an essential step in the classification of such manifolds, originally established by J. Kim in dimension $4$ and subsequently extended to dimensions $n\ge 5$. Our proof offers two notable advantages: it is shorter and does not require the use of any specialized moving frame.

A result of R. Hamilton asserts that any convex hypersurface in an Euclidian space with pinched second fundamental form must be compact. Partly inspired by this result, twenty years ago, in, Remark 3.1 on page 650, the author formulated a problem asking if a complete Riemannian manifold with positively pinched Ricci curvature must be compact. There are several recent progresses, which are all rigidity results concerning the flat metric except the special case for the steady solitons. In this note we provide a detailed alternate proof of Hamilton's result, in view of the recent proof via the mean curvature flow requiring additional assumptions and that the original argument by Hamilton does lack of complete details. The proof uses a result of the author in 1998 concerning quasi-conformal maps. The proof here allows a generalization as well. We dedicate this article to commemorate R. Hamilton, the creator of the Ricci flow, who also made fundamental contributions to many other geometric flows.

In this work, we consider a perturbation of an asymptotically conical gradient expanding Kähler-Ricci soliton metric $g$ in the same Kähler class. We demonstrate that, under suitable assumptions, the normalized Kähler-Ricci flow starting from the initial perturbed metric exists for all time and converges uniformly to an asymptotically conical gradient expanding Kähler-Ricci soliton metric $g_{\infty }$. Moreover, if the perturbed initial metric is asymptotic to $g$ at spatial infinity, then the limiting metric coincides with the original soliton, that is, $g_{\infty }=g$.

We study the Bergman metric and introduce the Bergman dual on Cartan-Hartogs (CH) domains. For a bounded domain D in C^n with Bergman kernel K_D, we define the Bergman dual of (D, g_D) as (D*, g_D*), where D* is the maximal domain on which the modified kernel K_D*(z, zbar) = K_D(z, -zbar) is positive, and g_D* is the Kahler metric obtained from K_D*. For a Cartan-Hartogs domain M_Omega, mu we prove the equivalence of: (i) M_Omega, mu is biholomorphic to the unit ball; (ii) its Bergman metric is a Kahler-Ricci soliton; (iii) after rescaling by a constant factor, the Bergman dual is finitely projectively induced. Conditions (i) and (ii) are Bergman-metric analogues of classical rigidity for Kahler-Einstein metrics (related to Yau's problem and Cheng's conjecture) and to recent rigidity for Kahler-Ricci solitons. Condition (iii) emphasizes the duality viewpoint, inspired by bounded symmetric domains and their compact duals. We also compare our results with other canonical metrics on CH domains, namely g_Omega, mu and hat g_Omega, mu, and discuss open problems about the maximal domain on which the Bergman dual is defined.

In this survey paper, we analyse and compare the recent curvature estimates for three types of $4$-dimensional gradient Ricci solitons, especially between Ricci shrinkers [58] and expanders [17]. In addition, we provide some new curvature estimates for $4$-dimensional gradient steady Ricci solitons, including the sharp curvature estimate $|Rm|\le CR$ for gradient steady Ricci solitons with positive Ricci curvature (see Theorem 1.1).

In this paper, we investigate curvature pinching phenomena in complete non-compact asymptotically conical gradient expanding Ricci solitons and establish several Hamilton-Ivey type curvature pinching estimates. These results are parallel to those known for shrinking and steady Ricci solitons. In particular, we prove a three-dimensional Hamilton-Ivey type curvature pinching theorem: any three-dimensional non-compact gradient Ricci expander, which is asymptotic to a cone with positive scalar curvature, must have positive sectional curvature. Furthermore, we formulate a general method and apply it to obtain analogues of several additional known generalized Hamilton-Ivey type curvature pinching results for ancient solutions. Among these is a curvature pinching estimate for four-dimensional asymptotically conical Ricci expanders with uniformly positive isotropic curvature, analogous to a result for four-dimensional gradient steady solitons due to Brendle [8].

We study the modified Ricci solitons as a new class of Einstein type metrics that contains both Ricci solitons and $n$-quasi-Einstein metrics. This class is closely related to the construction of the Ricci solitons that are realised as warped products. A modified Ricci soliton appears as part of a special solution of the modified Ricci-harmonic flow, which result a new characterization of $n$-quasi-Einstein metrics. We also study the modified Ricci almost solitons. In the spirit of the Lichnerowicz and Obata first eigenvalue theorems, we prove that in the class of compact Riemannian manifolds with constant scalar curvature the standard sphere with a structure of gradient modified Ricci almost soliton is rigid under some specific geometric conditions. Moreover, we display an example of modified Ricci-harmonic soliton.

We find new examples of steady gradient Ricci solitons with positive curvature operator in dimensions four and above. Utilising a procedure first introduced by Lai, we construct examples with $O(p)\times O(q)$ symmetry in dimension $p+q$ for any pair of integers $p,q\ge 2$ and discuss their asymptotic geometry.

We clarify the definition of polarized canonical radius in the study of Kahler Ricci flow.

This paper proves that there exists a non-trivial ancient solution to the Ricci flow emerging from the Taub-Bolt metric.

This paper shows for the first time the existence of a Ricci flow with surgery with local topology change \mathbbCP^2\setminus\{ pt\} \rightarrow \mathbbR^4. The post surgery flow converges to the Taub-NUT metric on \mathbbR^4 in infinite time.

Deep neural networks learn feature representations via complex geometric transformations of the input data manifold. Despite the models' empirical success across domains, our understanding of neural feature representations is still incomplete. In this work we investigate neural feature geometry through the lens of discrete geometry. Since the input data manifold is typically unobserved, we approximate it using geometric graphs that encode local similarity structure. We provide theoretical results on the evolution of these graphs during training, showing that nonlinear activations play a crucial role in shaping feature geometry in feedforward neural networks. Moreover, we discover that the geometric transformations resemble a discrete Ricci flow on these graphs, suggesting that neural feature geometry evolves analogous to Ricci flow. This connection is supported by experiments on over 20,000 feedforward neural networks trained on binary classification tasks across both synthetic and real-world datasets. We observe that the emergence of class separability corresponds to the emergence of community structure in the associated graph representations, which is known to relate to discrete Ricci flow dynamics. Building on these insights, we introduce a novel framework for locally evaluating geometric transformations through comparison with discrete Ricci flow dynamics. Our results suggest practical design principles, including a geometry-informed early-stopping heuristic and a criterion for selecting network depth.

In this paper, we study the evolution of metrics on finite trees under continuous-time Ricci flows based on the Lin-Lu-Yau version of Ollivier Ricci curvature. We analyze long-time dynamics of edge weights and curvatures, providing precise characterizations of their limiting behaviors. We prove that the Ricci flow converges to metric with zero curvature on edges whose normalized weights converge to positive values only if the tree is a caterpillar tree.

We investigate four-dimensional gradient shrinking Ricci solitons with positive modified sectional curvature. Our first main result shows that if the norm of the self-dual Weyl tensor and the scalar curvature satisfy a certain sharp pinching condition (closely related, in a precise sense, to that of Kähler metric), then the soliton is necessarily locally Kähler. We further obtain a characterization theorem and a weighted integral gap result for compact gradient shrinking Ricci solitons with bounded modified sectional curvature. In addition, we establish a Hitchin-Thorpe type inequality for compact four-dimensional Ricci solitons, providing new topological constraints on such manifolds.

Ricci curvature and Ricci flow have proven to be powerful tools for analyzing the geometry of discrete structures, particularly on undirected graphs, where they have been applied to tasks ranging from community detection to graph representation learning. However, their development on directed graphs remains limited, with Ricci flow being especially underexplored. In this work, we introduce a rigorous formulation of Ricci flow on directed weighted graphs, which evolves edge weights while preserving distances, and establish both the existence and uniqueness of its solutions. To capture the essence of asymmetry in directed networks and to enhance the capability of modeling more flexible structures, we incorporate a node-wise balancing factor that regulates between outflow and inflow. Building on the continuous Ricci flow evolution framework, we propose a discrete Ricci flow algorithm that is applicable to numerical computing. Numerical studies on various directed graph examples demonstrate the capacity of the proposed flow to reveal structural asymmetry and dynamic evolutions.

Lorantzian trans-Sasakian space form is a special type of space form in which the nature of even and odd dimensional space form both exist. Various curvature tensors with respect to Levi-Civita connection on the space form are derived in this paper. We have shown that if an odd-dimensional Lorentzian trans-Sasakian space form admits a hyperbolic Ricci soliton and hyperbolic conformal Ricci soliton then they will be $\eta$-Einstein. We also obtained the conditions for the solitons to be expanding, steady or shrinking. Finally, an example has been constructed which justifies the results obtained.

This paper attempts to construct monotonic entropy functionals for four-dimensional Lorentzian spacetime under physical boundary conditions, as an extension of Perelman's monotonic entropy functionals constructed for three-dimensional compact Riemannian manifolds. The monotonicity of these entropy functionals is utilized to prove the well-posedness of applying Ricci flow to four-dimensional Lorentzian spacetime for a long flow-time, particularly for the timelike modes which would seem blow up and ill-defined. The general idea is that the Ricci flow of a Lorentzian spacetime metric and the coupled conjugate heat flow of a density on the Lorentzian spacetime as a whole turns out to be the gradient flows of the monotonic functionals for a long flow-time, so the superficial "blow-up" in the individual Ricci flow system or the conjugate heat flow system contradicts the boundedness of the monotonic functionals within finite flow interval, which gives a semi-global control to the whole coupled system. The physical significance and applications of these monotonic entropy functionals in real gravitational systems are also discussed.

We first prove a uniform integral Laplace comparison result for the Kähler Ricci flow on Fano manifolds which depends only on the initial metric. As an application, using Cheeger-Colding theory and previous results by some of the authors, we give a direct and independent proof of the Hamilton-Tian conjecture on convergence of Kähler-Ricci flows, modulo a codimension 4 singular set. We also expounded on some existing literature on this conjecture.

In this article we prove an $\epsilon$-regularity theorem for non-collapsed Ricci flows, and use this to prove new estimates for singularity models of Fano Kähler-Ricci flows. In the course of our proof, we find a criterion for uniform convergence of solutions to the heat equation along a sequence of $\mathbb{F}$-converging Ricci flows, and apply this to new parabolic regularizations of some natural geometric quantities.

For all dimensions $n\ge 5$, let $(M,g,f)$ be a $n-$dimensional shrinking gradient Ricci soliton with strictly positive isotropic curvature (PIC). Suppose furthermore that ${\nabla }^{2}f$ is $2-$nonnegative and the curvature tensor is WPIC1 at some point $\bar{x}\in M$. Then $(M,g)$ must be a quotient of either $S^n$ or $S^{n-1}\times \mathbb{R}$. Our result partially extends the classification result for 4-dimensional PIC shrinking Ricci solitons established in [LNW16] to highter dimensions. Combining the pinching estimates deduced in [Chen24] we also extend the result in [CL23] to dimensions $n\ge 9$. Namely that a complete ancient solution to the Ricci flow of dimension $n\ge 9$ with uniformly PIC must be weakly PIC2.

We investigate the properties of the renormalisation group (RG) flow of two-dimensional sigma models with a generic metric coupling by utilising known results for the Ricci flow. We point out that on many occasions the RG flow develops singularities, due to strong coupling behaviour, before it reaches a UV or an IR fixed point. We illustrate our analysis with several examples. We give particular emphasis to type I singularities, where the length of the curvature of the sigma model target space grows at most as $|t-T{|}^{-1}$ as the flow parameter $t$ approaches the singularity at $T$. For these, the geometry near the singularity is described in terms of a shrinking Ricci soliton that exhibits a cosmological constant even though the original RG flow does not. Assuming that the spacetime satisfies an RG flow equation, we use the Ricci solitons to introduce a cosmological constant in a string theory setting. This can allow for different cosmological constants at different regions of spacetime. In particular, we point out how the de-Sitter space is a solution of the theory. We also raise the question on whether the techniques used to prove the geometrisation conjecture can be applied to prove the homogeneity and isotropy of the universe at large scales.

In this note, we give a diffeomorphism (to ${\mathbb{R}}^n$) criterion via long-time Ricci flow and show some applications. In particular, we provide an affirmative answer that the conclusion in [Manifolds with small curvature concentration, Ann. PDE, 2024] by Chan, Lee and the first named author and [Removing scalar curvature assumption for Ricci flow smoothing, Bull. Lond. Math. Soc., 2025] by A. Martens about manifolds with small curvature concentration can be improved to diffeomorphism in dimension $4$.

We prove that the four-dimensional blowdown shrinking Ricci soliton constructed by Feldman-Ilmanen-Knopf is strictly linearly stable in the sense of Cao-Hamilton-Ilmanen. This provides the first known example of a non-cylindrical linearly stable shrinking Ricci soliton. This offers new insights into the topological behavior of generic solutions to the Ricci flow in four dimensions: on top of reversing connected sums and handle surgeries, they should also undo complex blow-ups. The proof starts from an explicit description of the metric and develops a tensor harmonic analysis, adapted to its weighted Lichnerowicz Laplacian and based on its $U(2)$-invariance. It further exploits the Kähler structure of the blowdown shrinking soliton and insights from four-dimensional selfduality. The main difficulty is that the weighted Lichnerowicz Laplacian of the soliton admits a $9$-dimensional set of eigentensors associated with nonnegative eigenvalues. We show that they correspond to the Ricci tensor and gauge transformations.

We establish a local equivalence between toric steady Kähler-Ricci solitons and $A$-type toric generalized Kähler-Ricci solitons (GKRS). Under natural global conditions we show this equivalence extends to complete GKRS, yielding a general construction of new examples in all dimensions. We show that in four dimensions, all GKRS are either described by the generalized Kähler Gibbons-Hawking ansatz, or have split tangent bundle, or are $A$-type toric. This yields a local classification in four dimensions, together with a conjecturally exhaustive construction of complete symplectic-type examples.

We establish integral curvature estimates for complete gradient shrinking Sasaki-Ricci solitons. As an application, we show that any such soliton with harmonic Weyl tensor must be a finite quotient of a sphere. This result can be regarded as the Sasaki analogue of the work of Munteanu and Sesum [15] on Ricci solitons.

This paper studies minimal surface entropy (the exponential asymptotic growth of the number of minimal surfaces up to a given value of area) for negatively curved metrics on hyperbolic $3$-manifolds of finite volume, particularly its comparison to the hyperbolic minimal surface entropy in terms of sectional and scalar curvature. On one hand, for metrics that are bilipschitz equivalent to the hyperbolic metric and have sectional curvature bounded above by $-1$ and uniformly bounded below, we show that the entropy achieves its minimum if and only if the metric is hyperbolic. On the other hand, by analyzing the convergence rate of the Ricci flow toward the hyperbolic metric, we prove that among all metrics with scalar curvature bounded below by $-6$ and with non-positive sectional curvature on the cusps, the entropy is maximized at the hyperbolic metric, provided that it is infinitesimally rigid. Furthermore, if the metrics are uniformly $C^0$-close to the hyperbolic metric and asymptotically cusped, then the entropy associated with the Lebesgue measure is uniquely maximized at the hyperbolic metric.

On a hyperbolic 3-manifold of finite volume, we prove that if the initial metric is sufficiently close to the hyperbolic metric $h_0$, then the normalized Ricci-DeTurck flow exists for all time and converges exponentially fast to $h_0$ in a weighted Hölder norm. A key ingredient of our approach is the application of interpolation theory. Furthermore, this result is a valuable tool for investigating minimal surface entropy, which quantifies the growth rate of the number of closed minimal surfaces in terms of genus. We explore this in [17].

We prove that every nearly spherical, positively curved surface is the contractive, volume-preserving image of a round sphere. The proof combines three main tools: the Ricci flow on surfaces, the Kim-Milman construction, and a multiscale Bakry-Émery criterion.

Recent works of Guo-Phong-Song-Sturm established for compact Kähler manifolds (even for Kähler spaces of specific singularities) a variety of geometric estimates depending on an upper bound of $L^{1+\epsilon }$ or $L^1(\log L{)}^{n+\epsilon }$ norms of the volume density but not on any curvature bound, in which a key ingredient is a uniform integral estimate for Green's function. Motivated by their results and further applications, in this paper we shall prove an improved (nearly optimal) integral estimate for Green's function under $L^{1+\epsilon }$ volume density condition, and then apply it to obtain improved global geometric estimates. For instance, one of our results states that the $k$th eigenvalue of Laplacian operator ${\lambda }_k\ge c\cdot k^{\frac{1}{n}}(\log k{)}^{-3}$, where $n$ is the complex dimension of the Kähler manifold and $c$ depends on $n$ and $L^{1+\epsilon }$ norm of the volume density. Also, our results can be applied to the long-time or volume-noncollapsing finite-time Kähler-Ricci flow on compact Kähler manifolds and to a general Kähler family to further extend previous works of Guo-Phong-Song-Sturm, Guedj-Tô and Vu.

In this paper, we study the geometry and topology of complete gradient shrinking Sasaki-Ricci solitons. We first prove that they must be connected at infinity. This is a Sasaki analogue of gradient shrinking Kähler-Ricci solitons. Secondly, with the positive sectional curvature or positive transverse holomorphic bisectional curvature, we show that they must be compact. All results are served as a generalization of Perelman in dimension three, of Naber in dimension four, and of Munteanu-Wang in all dimensions, respectively.

In this paper, we study hypersurfaces in the product spaces ${\mathbb{Q}}_{\epsilon }^3\times \mathbb{R}$ for which the tangential component $T$ of the vector field $\frac{\partial }{\partial t}$ is a principal direction, where ${\mathbb{Q}}_{\epsilon }^3$ denotes the three-dimensional non-flat Riemannian space form with sectional curvature $\epsilon =\pm 1$, and $\frac{\partial }{\partial t}$ is the unit vector field tangent to the $\mathbb{R}$-factor. We obtain a local classification of hypersurfaces with three distinct principal curvatures satisfying specific functional relations. Then, we determine the necessary and sufficient conditions for such hypersurfaces to admit an almost Ricci soliton structure with potential vector field $T$. Finally, we prove that the only hypersurfaces admitting such solitons are rotational, by showing that the constructed examples with three distinct principal curvatures do not admit almost Ricci solitons.

The BRIDGES meeting in gauge theory, extremal structures, and stability was held June 2024 at l'Institut d'Études Scientifiques de Cargèse in Corsica, organized by Daniele Faenzi, Eveline Legendre, Eric Loubeau, and Henrique Sá Earp. The first week was a summer school consisting of four independent but related lecture series by Oscar García Prada, Spiro Karigiannis, Laurent Manivel, and Ruxandra Moraru. The present document consists of notes for the lecture series by Spiro Karigiannis on "Flows of geometric structures, especially ${\mathrm{G}}_2$-structures". Some assistance in the preparation of these notes by the author was provided by several participants of the summer school. See the Comments field for more information. The main theme is short time existence (STE) and uniqueness for geometric flows. We first introduce geometric structures on manifolds and geometric flows of such structures. We discuss some qualitative features of geometric flows, and consider the notions of strong and weak parabolicity. We focus on the Ricci flow, explaining carefully the DeTurck trick to establish short-time existence and uniqueness, an argument which we then extend to a general class of geometric flows of Riemannian metrics, previewing similar ideas for flows of ${\mathrm{G}}_2$-structures. Finally, we consider geometric flows of ${\mathrm{G}}_2$-structures. We review the basics of ${\mathrm{G}}_2$-geometry and survey several different geometric flows of ${\mathrm{G}}_2$-structures. In particular, we clarify in what sense STE results for the ${\mathrm{G}}_2$ Laplacian flow differ from STE results for other geometric flows. We conclude with a summary of some recent results by the author with Dwivedi and Gianniotis, including a classification of all possible heat-type flows of ${\mathrm{G}}_2$-structures, and a sufficient condition for such a flow to admit STE and uniqueness by a modified DeTurck trick.

The gravitational field equations in general relativity (GR) consist of a sophisticated system of nonlinear partial differential equations. Solving such equations in some generic off-diagonal forms is usually a hard analytic or numeric task. Physically important solutions in GR were constructed using a diagonal ansatz for metrics with a maximum of 4 independent coefficients. The Einstein equations can be solved in exact or parametric forms determined by some integration constants for corresponding assumptions on spherical or cylindrical spacetime symmetries. The anholonomic frame and connection deformation method allows us to construct generic off-diagonal solutions described by 6 independent coefficients of metrics depending, in general, on all spacetime coordinates. New types of exact and parametric solutions are determined by generating and integration functions and (effective) generating sources. They may describe vacuum gravitational and matter fields solitonic hierarchies; locally anisotropic polarizations of physical constants for black holes, wormholes, black toruses, or cosmological solutions; various types of off-diagonal deformations of horizons, etc. The additional degrees of freedom (related to off-diagonal coefficients) can be used to describe dark energy and dark matter configurations and elaborate locally anisotropic cosmological scenarios. In general, the generic off-diagonal solutions do not involve certain hypersurface or holographic configurations and can't be described in the framework of the Bekenstein-Hawking thermodynamic paradigm. We argue that generalizing the concept of G. Perelman's entropy for relativistic Ricci flows allows us to define and compute geometric thermodynamic variables for all possible classes of solutions in GR.

A Ricci soliton is a natural generalization of an Einstein metric. On a pseudo-Riemannian manifold (M, g), it is defined by : $LX g + \rho = λ g, where X is a smooth vector field on M , LX denotes the Lie derivative in the direction of X, \rho is the Ricci tensor, and λ is a real constant. In this paper, we establish the existence of non-trivial Ricci solitons on a family of three-dimensional Lorentzian Walker manifolds.

Weak metric structures, introduced by Rovenski and Wolak in 2022, extend Yano's $f$-structure and almost contact metric structure. In this paper, we investigate curvature phenomena of weak almost $\mathcal{S}$-manifolds (w.a.$\mathcal{S}$-manifolds) focusing on the $f$-$(\kappa ,\mu )$-nullity condition and its special case $R_{X,Y}\xi =0$. We establish several results that generalize known rigidity theorems for almost $\mathcal{S}$-manifolds. First, using the partial Ricci flow, we obtain dynamical characterizations of $\mathcal{S}$-manifolds: starting from a w.a.$\mathcal{S}$-structure satisfying the curvature condition of $\mathcal{S}$-manifolds or the $f$-$(1,\mu )$-nullity condition, the flow evolves the structure exponentially fast toward an $\mathcal{S}$-structure. This extends results of Cappelletti Montano and Di Terlizzi to the weak metric setting. Next, we identify conditions under which a w.a.$\mathcal{S}$-manifold admits a bi-Legendrian structure with totally geodesic foliations. Finally, for w.a.$\mathcal{S}$-manifolds with $\kappa =\mu =0$, we prove a splitting theorem in which one factor is flat, generalizing classical results for almost $\mathcal{S}$-geometry. These findings have consequences for the theory of Sasakian and $\mathcal{S}$- manifolds, the geometry of bi-Legendrian structures, and the behavior of weak metric contact manifolds under curvature constraints.

This article deals with the problems of preserving the Ricci curvature positivity on homogeneous spaces under the normalized Ricci flow (NRF). We found out infinitely many generalized Wallach spaces (GWS) on which the positivity of the Ricci curvature of metrics is preserved when evolved by the NRF. Analogously, the number of GWS is infinite as well, when the positivity of the Ricci curvature can be lost. We also obtain some refinements to our previous results devoted to the case of coincided parameters. A series of examples is discussed.

We solve a general class of free boundary Monge-Ampère equations given by $$\det D^{2}u=\lambda \frac{f(-u)}{g(u^{\star })h(\nabla u)}{\chi }_{\{u<0\}}\text{ in }{\mathbb{R}}^n,\nabla u({\mathbb{R}}^n)=P$$ where $P$ is a bounded convex set containing the origin, and $h>0$ on $P$. We consider applications to optimal transport with degenerate densities, Monge-Ampère eigenvalue problems, and geometric problems including a hemispherical Minkowski problem and free boundary Kähler-Ricci solitons on toric Fano manifolds.

Graph Ricci curvature is crucial as it geometrically quantifies network structure. It pinpoints bottlenecks via negative curvature, identifies cohesive communities with positive curvature, and highlights robust hubs. This guides network analysis, resilience assessment, flow optimization, and effective algorithm design. In this paper, we derived upper and lower bounds for the weights along several kinds of discrete Ricci curvature flows. As an application, we utilized discrete Ricci curvature flows to detect the core subgraph of a finite undirected graph. The novelty of this work has two aspects. Firstly, along the Ricci curvature flow, the bounds for weights determine the minimum number of iterations required to ensure weights remain between two prescribed positive constants. In particular, for any fixed graph, we conclude weights can not overflow and can not be treated as zero, as long as the iteration does not exceed a certain number of times; Secondly, it demonstrates that our Ricci curvature flow method for identifying core subgraphs outperforms prior approaches, such as page rank, degree centrality, betweenness centrality and closeness centrality. The codes for our algorithms are available at https://github.com/12tangze12/core-detection-via-Ricci-flow.

Riemannian Einstein solvmanifolds can be described in terms of nilsolitons, namely nilpotent Lie groups endowed with a left-invariant Ricci soliton metric. This characterization does not extend to indefinite metrics; nonetheless, nilsolitons can be defined and used to construct Einstein solvmanifolds of a higher dimension in any signature. An Einstein solvmanifold obtained by this construction turns out to satisfy the pseudo-Iwasawa condition, meaning that its Lie algebra splits as the orthogonal sum of a nilpotent ideal and an abelian subalgebra, the latter acting by symmetric derivations. In this paper we construct a family of pseudo-Iwasawa solvmanifolds admitting a Killing spinor in any dimension and signature and prove that all pseudo-Iwasawa solvmanifolds admitting a Killing spinor, invariant or not, belong to this family. If in addition the metric is Einstein, we show that the only possibility is the hyperbolic half-space. As a byproduct, we prove that the only homogeneous Riemannian manifold admitting a Killing spinor with imaginary Killing constant is hyperbolic space.

We prove a universal embedding theorem for flag manifolds: every flag manifold admits a holomorphic isometric embedding into an irreducible classical flag manifold. This result generalizes the classical celebrated embedding theorems of Takeuchi [30] and Nakagawa-Takagi [27]. Using this embedding, we establish new rigidity phenomena for holomorphic isometries between homogeneous Kähler manifolds. As a first immediate consequence we show the triviality of a Kähler-Ricci soliton submanifod of $C\times \Omega$, where $C$ is a flag manifold and $\Omega$ is a homogeneous bounded domain. Secondly, we show that no weak-relative relationship can occur among the fundamental classes of homogeneous Kähler manifolds: flat spaces, flag manifolds, and homogeneous bounded domains. Two Kähler manifolds are said to be weak relatives if they share, up to local isometry, a common Kähler submanifold of complex dimension at least two. Our main result precisely shows that if $E$ is (possibly indefinite) flat, $C$ is a flag manifold, and $\Omega$ is a homogeneous bounded domain, then: $E$ is not weak relative to $C\times \Omega$; $C$ is not weak relative to $E\times \Omega$; $\Omega$ is not weak relative to $E\times C$. This extends, in two independent directions, the rigidity theorem of Loi-Mossa [22]: we pass from relatives to the more flexible notion of weak relatives and dispense with the earlier "special" restriction on the flag-manifold factor. This result also unifies previous rigidity results from the literature, e.g., [5, 6, 7, 9, 12, 13, 32].

The present work constitutes the third installment in a series of investigations devoted to discrete conformal structures on surfaces with boundary. In our preceding works, we established, respectively, a classification of these discrete conformal structures and results on their rigidity and existence. Building on this foundation, the present work focuses on the deformation theory of discrete conformal structures on surfaces with boundary. Specifically, we introduce the combinatorial Ricci flow and the combinatorial Calabi flow, and establish the longtime existence and global convergence of solutions to these combinatorial curvature flows. These results yield effective algorithms for finding discrete hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.

We prove the dynamical Alekseevski conjecture in dimension five. We also provide a detailed analysis of the homogeneous Ricci flows on $SO(3)⋉{\mathbb{R}}^3/SO(2)$ and $SL(2,\mathbb{C})/U(1)$.

In this work, we study gradient solitons to general geometric flows. Our approach is to understand what assumptions need to be made about a flow in order to extend results about Ricci solitons. In this direction, we identify an identity, first exploited in the pioneering work of Richard Hamilton in the case of Ricci solitons, which we call Hamilton's identity. We show that a version of this identity for an arbitrary geometric flow allows one to recover results about rigidity, the growth of the potential function, volume growth and the Omori-Yau maximum principle that have been proven for gradient Ricci solitons.

We study the properties of LP-Sasakian manifolds endowed with generalized Ricci solitons associated to the general connection. Finally, the existence of such solitons on a 4-dimensional LP-Sasakian manifold is proved by constructing a non-trivial example.

We show that for a broad family of noncompact homogeneous Riemannian manifolds, the corresponding homogeneous Ricci flow solutions have finite extinction time, thereby confirming the dynamical Alekseevskii conjecture for these spaces. As an application, we prove that on such homogeneous manifolds $G/H$, the space of all $G$-invariant positive scalar curvature metrics is contractible.

Optimal transport plays a major role in the study of manifolds with Ricci curvature bounded below. Some results in this setting have been extended to super Ricci flows, revealing a unified approach to analysis on Ricci nonnegative manifolds and Ricci flows. However we observe that the monotonicity of Perelman's functionals ($\mathcal{F}$, $\mathcal{W}$, reduced volume), which hold true for Ricci flows and Ricci nonnegative manifolds, cannot be strictly generalized to super Ricci flows. In 2010 Buzano introduced a condition which still generalizes Ricci flows and Ricci nonnegative manifolds, and on which Perelman's monotonicities do hold. We provide characterizations of this condition using optimal transport and understand it heuristically as Ricci nonnegativity of the space-time. This interpretation is consistent with its equivalence to Ricci nonnegativity on Perelman's infinite dimensional manifold. More precisely, we prove that for smooth evolutions of Riemannian manifolds, this condition is equivalent to a Bochner inequality (resembling Perelman's Harnack inequality but for the forward heat flow), a gradient estimate for the heat flow, a Wasserstein contraction along the adjoint heat flow, the convexity of a modified entropy along Wasserstein geodesics, and an Evolutionary Variational Inequality (EVI). The optimal transport statements use Perelman's $L$ distance as cost, as first studied on Ricci flows by Topping and by Lott. We also consider dimensionally improved and weighted versions of these conditions. The dimensional Bochner inequality and all gradient estimates for the forward heat equation, along with the EVIs, appear to be new even for general Ricci flows, and are related to the Hamiltonian perspective on the $L$ distance. Most of our proofs do not use tensor calculus or Jacobi fields, suggesting the possibility of future extensions to more singular settings.

We present a comprehensive analysis of Bredon's trick, a powerful local-to-global extension principle with broad applications across differential geometry and computational topology. Our main contributions include: (1) novel applications to stratified pseudomanifolds via Verona cohomology with explicit verification of axiomatic conditions; (2) new frameworks for Ricci flow singularity analysis using local curvature concentration; (3) stability theorems for persistent homology in distributed computational settings; and (4) rigorous applications to medical imaging and neural network topology. By systematically developing the theoretical foundations and providing concrete implementations, this work establishes Bredon's trick as a unifying framework for modern local-to-global arguments in geometric analysis and applied topology.

We revisit the recent theory of Sun-Zhang on general Fano fibration (germs) which emerged from the study of non-compact Kahler-Ricci soliton metrics, primarily from an algebro-geometric perspective. In addition to reviewing the existing framework, we present new results, conjectures, and remarks. These include methods for computing weighted volumes via (restricted) volumes, Laplace transforms, and incomplete Gamma-functions, and a conjectural algebro-geometric construction ("bubbling") of Fano fibration with asymptotically conical base from degenerating Fano fibration.

Motivated by the long-time behavior of Ricci flows that collapse with bounded curvature, we study expanding Ricci solitons with nilpotent symmetry on vector bundles over a closed manifold. We prove that, under mild assumptions that are satisfied by Ricci flow limits, the equations dimension-reduce to the so-called twisted harmonic-Einstein equations. When the base is a surface, we establish a correspondence between solutions of the latter and a class of G-Higgs bundles. This allows us to produce infinite families of new examples that are not locally homogeneous, and in particular to obtain a complete description in dimension 4. We also show that all our examples admit Einstein one-dimensional extensions.

Recent interest among geometers in $f$-structures of K. Yano is due to the study of topology and dynamics of contact foliations and generalized A. Weinstein conjectures. Weak metric $f$-structures, introduced by the author and R. Wolak as a generalization of Hermitian structure, as well as $f$-structure allow for a fresh perspective on the classical theory. An important case of such manifolds, which is locally a twisted product, is a weak $\beta f$-Kenmotsu manifold defined as a generalization of K. Kenmotsu's concept. In this paper, the concept of the $\ast$-Ricci tensor of S. Tashibana is adapted to weak metric $f$-manifolds, the interaction of $\ast$-$\eta$-Ricci soliton with the weak $\beta f$-Kenmotsu structure is studied and new characteristics of $\eta$-Einstein metrics are obtained.

In his seminal work, Rivin characterized finite ideal polyhedra in three-dimensional hyperbolic space. However, the characterization of infinite ideal polyhedra, as proposed by Rivin, has remained a long-standing open problem. In this paper, we introduce the combinatorial Ricci flow for infinite ideal circle patterns, a discrete analogue of Ricci flow on non-compact Riemannian manifolds, and prove a characterization of such circle patterns under certain combinatorial conditions. Our results provide affirmative solutions to Rivin's problem.

In this paper, we establish Li-Yau-type and Hamilton-type estimates for positive solutions to the heat equation associated with the generalized Ricci flow, under a less stringent curvature condition. Compared with [25] and [35], these estimates generalize the results in Ricci flow to this new flow under the weaker Ricci curvature bounded assumption. As an application, we derive the Harnack-type inequalities in spacetime and find the monotonicity of one parabolic frequency for positive solutions of the heat equation under bounded Ricci curvature.

This paper studies a complete gradient Ricci soliton with an isoparametric potential function. Our first theorem asserts that, for the steady case, there is a critical level set of codimension greater than one. This is consistent with construction of cohomogeneity one models with singular orbits. There is a partial result for the shrinking case. We also study a particular ansatz of popular interest and obtain asymptotic behaviors.

Suppose $(M,g,f)$ is a 5-dimensional complete shrinking gradient Ricci soliton with $R=1$. If it has bounded curvature, we prove that it is a finite quotient of ${\mathbb{R}}^3\times {\mathbb{S}}^2$.

In this paper we introduce the branched $\alpha$-flows on closed surfaces with Euler characteristic $\chi \le 0$. Based on the strict convexity of the branched $\alpha$-potentials, we establish the long time existence and convergence of the solutions to the branched $\alpha$-flows, which generalizes Ge and Xu's main results on the $\alpha$-flows. In addtion, we study the prescribed curvature problems under the relaxed precondition $\chi (M)\in \mathbb{Z}$ via alternative $\alpha$-flows, establishing admissibility conditions for prescribed curvatures and their exponential convergence to target metrics.

We establish a short-time existence theory for complete Ricci flows under scaling-invariant curvature bounds, starting from rotationally symmetric metrics on ${\mathbb{R}}^{n+1}$ that are noncollapsed at infinity, without assuming bounded curvature. As a consequence, we construct a complete Ricci flow solution coming out of a rotationally symmetric metric, which has a cone-like singularity at the origin and no minimal hypersphere centered at the origin, using an approximation method.

In this paper, we explore a new class of integrable sigma models, which we refer to as the "dual regime" of Yang-Baxter (YB) deformed $\mathrm{O}(2N)$ sigma models. This dual regime manifests itself in the conformal perturbation approach. Namely, it is well known that conventional YB-deformed $\mathrm{O}(N)$ sigma models are described in the UV by a collection of free bosonic fields perturbed by some relevant operators. The holomorphic parts of these operators play the role of screening operators which define certain integrable systems in the free theory. All of these integrable systems depend on a continuous parameter $b$, which parametrizes the central charge, and are known to possess the duality under $b^2⟷-1-b^2$. Although $\mathrm{O}(2N+1)$ integrable systems are self-dual, $\mathrm{O}(2N)$ systems are not. In particular, the $\mathrm{O}(2N)$ integrable systems provide new perturbations of the sigma model type. We identify the corresponding one-loop metric and $B-$field and show that they solve the generalized Ricci flow equation.

We establish a symmetry principle for asymptotically cylindrical steady gradient Ricci solitons (GRSs) and asymptotically conical expanding GRSs with homogeneous links. Using this, we show that the Bryant steady soliton is the unique asymptotically cylindrical steady GRS that has a round spherical link and satisfies a particular quantitative rigidity condition. A similar characterization is proved for Bryant's expanding solitons. Finally, we establish a global symmetry result for GRSs which exhibit the aforementioned asymptotics with quotient-Berger sphere asymptotic links.

In this paper, we investigate the prescribed total geodesic curvature problem for generalized circle packing metrics in hyperbolic background geometry on surfaces with infinite cellular decompositions. To address this problem, we introduce a prescribed curvature flow-a discrete analogue of the Ricci flow on noncompact surfaces-specifically adapted to the setting of infinite cellular decompositions. We establish the well-posedness of the flow and prove two convergence results under certain conditions. Our approach resolves the prescribed total geodesic curvature problem for a broad class of surfaces with infinite cellular decompositions, yielding, in certain cases, smooth hyperbolic surfaces of infinite topological type with geodesic boundaries or cusps. Moreover, the proposed flow provides a method for constructing hyperbolic metrics from appropriate initial data.

Let (X, g, J, f ) be a non-compact gradient shrinking Kahler-Ricci soliton. We prove that if the scalar curvature of X satisfies a mild assumption, then OP (X), the ring of holomorphic functions with polynomial growth on X, is finitely generated. This gives a partial confirmation to a conjecture of Munteanu and Wang (cf.[MW14]).

In this paper we introduce a new functional on the space of $G_2$-structures which we call the $G_2$-Hilbert functional. It is uniquely determined by a few basic principles inspired by the Einstein-Hilbert functional in Riemannian Geometry, and it has similar variational behaviour with it. For instance, torsion-free and nearly $G_2$-structures are saddle critical points of the volume-normalized $G_2$-Hilbert functional. This allows us to uniquely distinguish two new flows of $G_2$-structures, which can be considered as analogues of the Ricci flow in $G_2$-geometry.

Since the fundamental work of Chow-Luo, Ge et al., the combinatorial curvature flow methods became a basic technique in the study of circle pattern theory. In this paper, we investigate the combinatorial Ricci flow with prescribed total geodesic curvatures in spherical background geometry. For infinite cellular decompositions, we establish the existence of a solution to the flow equation for all time. Furthermore, under an additional condition, we prove that the solution converges as time tends to infinity. To the best of our knowledge, this is the first study of an infinite combinatorial curvature flow in spherical background geometry.

In this paper, we extend Perelman's $W$-entropy formula and the concavity of the Shannon entropy power from smooth Ricci flow to super Ricci flows on metric measure spaces. Moreover, we prove the Li-Yau-Hamilton-Perelman Harnack inequality on super Ricci flows. As a significant application, we prove the equivalence between the volume non-local collapsing property and the lower boundedness of the $W$-entropy on RCD$(0,N)$ spaces. Finally, we use the $W$-entropy to study the logarithmic Sobolev inequality with optimal constant on super Ricci flows on metric measure spaces.

In this paper, we characterize the Ricci soliton equations on the Poincaré upper half plane . First we classify all Ricci soliton and Ricci Bourguignon soliton in the half plane of Poincaré and after we generalize those equations in ${\mathbb{H}}^n$. We obtain some nice properties of the soliton about their geodesic flows.

This note corrects an error in the proof of Proposition 13 in arXiv:1903.09181 and simultaneously establishes a more general result. We prove that if $M$ is a compact connected oriented $4$-manifold with connected boundary $\partial M$, and if an unbounded number of disjoint copies of $M$ embed topologically and locally flatly in the interior of a compact $4$-manifold $N,$ then $\mathrm{Tor}{H}_1(\partial M;\mathbb{Z})$ is a direct double, i.e., $\mathrm{Tor}{H}_1(\partial M;\mathbb{Z})\cong A\oplus A$, with the linking pairing vanishing identically on the first summand, i.e., the linking pairing is split metabolic. This partially generalizes Hantzsche's theorem stating that the linking pairing for a closed $3$-manifold that embeds in $S^4$ is hyperbolic.

We study two different natural notions of singular Kähler-Einstein metrics on normal complex varieties. In the setting of singular Ricci flat Kähler cone metrics that arise as non-collapsed limits of sequences of Kähler-Einstein metrics or Kähler-Ricci flows, we show that an a priori weaker notion is equivalent to the stronger one introduced by Eyssidieux-Guedj-Zeriahi, and in particular the underlying variety has log terminal singularities in this case. Our method applies to more general singular Kähler-Einstein spaces as well, assuming that they define RCD spaces.

In this paper, we begin by introducing Clairaut Riemannian warped product maps and establish the condition under which a regular curve becomes a geodesic. We obtain the conditions for a Riemannian warped product map to be Clairaut Riemannian warped product map followed by Ricci curvature. Further, we study the Ricci soliton structure on a Riemannian warped product manifold using curvature tensor. We examine the Bochner type formulae for Clairaut Riemannian warped product map and construct a supporting example. Furthermore, we extend the study to introduce and examine some geometric aspects of conformal Riemannian warped product maps. We derive the integral formula for scalar curvature of conformal Riemannian warped product map. Finally, we construct an example for conformal Riemannian warped product map.

We study the uniqueness problem for the Kähler-Ricci flow with a conical initial condition. Given a complete gradient expanding Kähler-Ricci soliton on a non compact manifold with quadratic curvature decay, including its derivatives, we establish that any complete solution to the Kahler-Ricci flow emerging from the soliton's tangent cone at infinity–appearing as a Kähler cone–must coincide with the forward self-similar Kähler-Ricci flow associated with the soliton, provided certain conditions hold. Specifically, if its Kähler form remains in the same cohomology class as that of the soliton's self-similar Kähler-Ricci flow, its full Riemann curvature operator is bounded for each fixed positive time, its Ricci curvature is bounded from above by A/t, its scalar curvature is bounded from below by -A/t, and it shares a same Killing vector field with the soliton metric. This paper gives a partial answer to a question in paper of Feldman-Ilmanen-Knopf, and generalizes the earlier work of Conlon-Deruelle and the work of Conlon-Deruelle-Sun.

Introducing a moment map whose zero locus is the group of symplectomorphisms of the real four-dimensional torus, we exhibit a gradient flow that can be made into a strictly parabolic flow by mean of a DeTurck trick (famously known for its use in the study of the Ricci flow), showing the local existence and regularity for the solutions of this flow and hence showing that the group of symplectomorphisms of the real four-dimensional torus is locally contractible. This work follows the ideas introduced by Yann Rollin in [3], even though the moment map picture comes from different considerations.

The $S_8$ parameter, which quantifies the amplitude of matter fluctuations on scales of $8h^{-1}$ Mpc, has been a source of tension between weak lensing surveys (e.g. KiDS, DES, HSC) and the Planck Cosmic Microwave Background (CMB) measurements. This discrepancy challenges the standard $\Lambda$CDM model and has become one of the most significant tensions in modern cosmology. The ${\Lambda }^{\prime }$ model offers a potential resolution by introducing modifications to the cosmic growth history through alterations to the gravitational sector. The alterations involve including a Ricci soliton into Einstein's field equations which introduce a time dependent factor yielding a time varying cosmological constant ${\Lambda }^{\prime }=(1-\alpha (t{)}^2{\Lambda }_{DE}\frac{{\rho }_g}{{\rho }_{DE}}$ and subsequently the evolution of the cosmos. The Ricci soliton is sourced from gravitational energy density. In this study we analyze results from six surveys and compare the results for $w_a$ and $w_0$ with the ${\Lambda }^{\prime }$ model. We also find ${\sigma }_8={0.750}_{-0.020}^{+0.020}$ , $S_8=0.788$. These values are closer to some low $S_8$ measurements from weak lensing surveys (e.g DES, KiDS), which report $S_8\approx 0.76-0.78$, suggesting that the model may alleviate the $S_8$ tension. High values of $\alpha (t)$ in the late universe are the cause of suppressed structure formation and low values of ${\Lambda }^{\prime }$. The late universe in the ${\Lambda }^{\prime }$ model is effectively or apparently 5-10% younger than in $\Lambda$CDM which translates to $H_0={72.734}_{-1.687}^{+1.687}$ km/s/Mpc, which is in agreement with late universe probes. ${\Lambda }^{\prime }$ is classified under the dynamical dark energy models, however unlike alternatives, it does not invoke exotic particles nor phantom energy.

In this paper, we study gradient Ricci soitons on smooth orbifolds. We prove that the scalar curvature of a complete shrinking or steady gradient Ricci soliton on an orbifold is nonnegative. We also show that a complete $\kappa$-noncollapsed steady gradient Ricci soliton on a Riemannian orbifold with positive curvature operator, compact singularity and linear curvature decay must be a finite quotient of the Bryant soliton. Finally, we show that a complete steady gradient Ricci soliton on a Riemannian orbifold with positive sectional curvature must be a finite quotient of the Bryant soliton if it is asymptotically quotient cylindrical.

In this work, we discuss the stability of the pluriclosed flow and generalized Ricci flow. We proved that if the second variation of generalized Einstein–Hilbert functional is nonpositive and the infinitesimal deformations are integrable, the flow is dynamically stable. Moreover, we prove that the pluriclosed steady solitons are dynamically stable when the first Chern class vanishes.

In this paper, we study the $t$-Gauduchon Ricci-flat condition under the Chern-Ricci flow. In this setting, we provide examples of Chern-Ricci flow on compact non-Kähler Calabi-Yau manifolds which do not preserve the $t$-Gauduchon Ricci-flat condition for $t<1$. The approach presented generalizes some previous constructions on Hopf manifolds. Also, we provide non-trivial new examples of balanced non-pluriclosed solution to the pluriclosed flow on non-Kähler manifolds. Further, we describe the limiting behavior, in the Gromov-Hausdorff sense, of geometric flows of Hermitian metrics (including the Chern-Ricci flow and the pluriclosed flow) on certain principal torus bundles over flag manifolds. In this last setting, we describe explicitly the Gromov-Hausdorff limit of the pluriclosed flow on principal $T^2$-bundles over the Fano threefold $\mathbb{P}(T_{{\mathbb{P}}^2})$.

Guo and Luo introduced generalized circle patterns on surfaces and proved their rigidity. In this paper, we prove the existence of Guo-Luo's generalized circle patterns with prescribed generalized intersection angles on surfaces with cusps, which partially answers a question raised by Guo-Luo and generalizes Bobenko-Springborn's hyperbolic circle patterns on closed surfaces to generalized hyperbolic circle patterns on surfaces with cusps. We further introduce the combinatorial Ricci flow and combinatorial Calabi flow for generalized circle patterns on surfaces with cusps, and prove the longtime existence and convergence of the solutions for these combinatorial curvature flows.

We show that the Liouville entropy of the geodesic flow of a closed surface of non-constant negative curvature is eventually strictly increasing along the normalized Ricci flow (NRF). More precisely, we obtain a new expression for the derivative of the Liouville entropy along an arbitrary conformal deformation in dimension 2, and we prove it is positive in the direction of the NRF for 1/6-pinched metrics. This partially answers a question of Manning from 2004. In addition, we show that the mean root curvature, a purely geometric quantity which is a lower bound for the Liouville entropy, is strictly increasing along the NRF starting from any metric of non-constant negative curvature.

In this paper, we study the collpasing Kähler-Ricci flow on Hirzebruch surfaces, which develops finite time singularities. We show that any tangent flow based at a point in the singular time slice is the Kähler-Ricci flow associated with a nonflat gradient Kähler-Ricci shrinker with finitely many orbifold singularities .

In this paper, we introduce combinatorial Ricci flows (CRFs in short) in Euclidean and hyperbolic background geometries on infinite triangulations of the open disk, which are discrete analogs of Ricci flows on simply connected open surfaces. We establish well-posedness results, the existence and the uniqueness, of CRFs in both Euclidean and hyperbolic background geometries. Moreover, we prove convergence results of CRFs, which indicate a uniformization theorem for CRFs on infinite disk triangulations. As an application, we prove an existence result of circle-packing metrics with infinite prescribed cone angles in hyperbolic background geometry. To our knowledge, these are the first results of CRFs on infinite triangulations.

In this paper, we prove that any compact 2-sided smooth stable minimal hypersurface in a shrinking gradient Ricci soliton $(M^n,g,f)$ with scalar curvature $R\ge (n-1)\lambda$ must have vanished second fundamental form and vanished normal Ricci curvature. For shrinking gradient Ricci solitons with scalar curvature $R\ge (n-1)\lambda$, the existence of an area-minimizing hypersurface would imply that $M$ is splitting.

For an ancient Ricci flow asymptotic to a compact integrable shrinker, or a Ricci flow developing a finite-time singularity modelled on the shrinker, we establish the long-time existence of a harmonic map heat flow between the Ricci flow and the shrinker for all times. This provides a global parabolic gauge for the Ricci flow and implies the uniqueness of the tangent flow without modulo any diffeomorphisms. We present two main applications: First, we construct and classify all ancient Ricci flows asymptotic to any compact integrable shrinker, showing that they converge exponentially. Second, we obtain the optimal convergence rate at singularities modelled on the shrinker, characterized by the first negative eigenvalue of the stability operator for the entropy. In particular, we show that any Ricci flow developing a round ${\mathbb{S}}^n$ singularity converges at least at the rate $(-t{)}^{\frac{n+1}{n-1}}$.

In this work, we prove uniqueness for complete non-compact Ricci flow with scaling invariant curvature bound. This generalizes the earlier work of Chen-Zhu, Kotschwar and covers most of the example of Ricci flows with unbounded curvature. In dimension three, we use it to show that complete Ricci flow starting from uniformly non-collapsed, non-negatively curved manifold is unique, extending the strong uniqueness Theorem of Chen. This is based on solving Ricci-harmonic map heat flow in unbounded curvature background.

This paper establishes a unified framework integrating geometric flows with deep learning through three fundamental innovations. First, we propose a thermodynamically coupled Ricci flow that dynamically adapts parameter space geometry to loss landscape topology, formally proved to preserve isometric knowledge embedding (Theorem ). Second, we derive explicit phase transition thresholds and critical learning rates (Theorem ) through curvature blowup analysis, enabling automated singularity resolution via geometric surgery (Lemma ). Third, we establish an AdS/CFT-type holographic duality (Theorem ) between neural networks and conformal field theories, providing entanglement entropy bounds for regularization design. Experiments demonstrate 2.1$\times$ convergence acceleration and 63% topological simplification while maintaining $\mathcal{O}(N\log N)$ complexity, outperforming Riemannian baselines by 15.2% in few-shot accuracy. Theoretically, we prove exponential stability (Theorem ) through a new Lyapunov function combining Perelman entropy with Wasserstein gradient flows, fundamentally advancing geometric deep learning.

Thanks to the ambitious project initiated by Catino, Mastrolia, Monticelli and Rigoli, which aims to provide a unified viewpoint for various geometric solitons, many classes, including Ricci solitons, Yamabe solitons, $k$-Yamabe solitons, quasi-Yamabe solitons, and conformal solitons, can now be studied under a unified framework known as Einstein-type manifolds. Einstein-type manifolds are characterized by four constants, denoted by $\alpha ,\beta ,\mu$ and $\rho$. In this paper, we completely classify all non-trivial, complete gradient Einstein-type Kähler manifolds with $\alpha =0$. As a corollary, rotational symmetry for many classes is obtained. In particular, we show that any non-trivial complete gradient quasi-Yamabe soliton on Kähler manifolds is rotationally symmetric.

Weak almost contact metric manifolds (i.e., the complex structure is replaced by a nonsingular skew-symmetric tensor), defined by the author and R. Wolak, allow a new look at the classical theory and find novel applications. An important case of these manifolds, which is locally a twisted product, is a weak $\beta$-Kenmotsu manifold defined by the author and D.S. Patra. In the paper, the concept of the $\ast$-Ricci tensor is adapted to weak almost contact manifolds, the interaction of the $\ast$-$\eta$-Ricci soliton with the weak $\beta$-Kenmotsu structure (with $\beta =const$) is studied and new characteristics of Einstein metrics are obtained.

In this survey, we consider various analytic problems related to the geometry of the Chern connection on Hermitian manifolds, such as the existence of metrics with constant Chern-scalar curvature, generalizations of the Kähler-Einstein condition to the non-Kähler setting, and the convergence of the Chern-Ricci flow on compact complex surfaces.

Simply-connected four-dimensional gradient Ricci solitons that are invariant under a compact cohomogeneity one group action have been studied extensively. However, the special case where the group is $SU(2)$ (the smallest possible example) has received comparatively little attention. The purpose of this article is to give a comprehensive study of simply-connected $SU(2)$-invariant expanding and shrinking cohomogeneity one gradient Ricci solitons. The first result is the construction of new 3-parameter families of complete $SU(2)$-invariant asymptotically conical expanding gradient Ricci solitons. New shrinking Kähler $U(2)$-invariant gradient Ricci solitons in dimension 4 with orbifold singularities are also constructed, leading to a classification of such metrics when the base space of the orbifold is a simply-connected smooth manifold. Finally, we highlight numerical evidence that all the compact cohomogeneity one shrinking gradient Ricci solitons are known.

In this work, we construct several sequences of metrics on sphere with different limiting behaviors. By combining with the work of Deruelle, we use it and the localized maximum principle to construct various examples of expanding gradient Ricci solitons with positive curvature and exotic curvature decay. This answers a question proposed by Chow-Lu-Ni and also a question by Cao-Liu, respectively.

Regarding Ricci flow as a dynamical system, we derive sufficient conditions for noncompact stationary (Ricci-flat) solutions to possess infinite-dimensional unstable manifolds, and provide examples satisfying those criteria that have uncountably many unstable perturbations.

Thurston's triangulation conjecture asserts that every hyperbolic 3-manifold admits a geometric triangulation into hyper-ideal hyperbolic tetrahedra. So far, this conjecture had only been proven for a few special 3-manifolds. In this article, we confirm this conjecture for a class of 3-manifolds. To be precise, let $M$ be an oriented compact 3-manifold with boundary, no component of which is a 2-sphere, and $\mathcal{T}$ is an ideal triangulation of $M$. If $\mathcal{T}$ satisfies properly gluing condition, and the valence is at least 6 at each ideal edge and 11 at each hyper-ideal edge, then $M$ admits an unique complete hyperbolic metric with totally geodesic boundary, so that $\mathcal{T}$ is isotopic to a geometric ideal triangulation of $M$. We use analytical tools such as combinatorial Ricci flow (CRF, abbr.) to derive the conclusions. There are intrinsic difficulties in dealing with CRF. First, the CRF may collapse in a finite time, second, most of the smooth curvature flow methods are no longer applicable since there is no local coordinates in $\mathcal{T}$, and third, the evolution of CRF is affected by certain combinatorial obstacles in addition to topology. To this end, we introduce the ideas as "extending CRF", "tetrahedral comparison principles", and "control CRF with edge valence" to solve the above difficulties. In addition, the presence of torus boundary adds substantial difficulties in this article, which we have solved by introducing the properly gluing conditions on $\mathcal{T}$ and reducing the ECRF to a flow relatively easy to handle.

We show the existence of a solution to the Ricci flow with a compact length space of bounded curvature, i.e., a space that has curvature bounded above and below in the sense of Alexandrov, as its initial condition. We show that this flow converges in the $C^{1,\alpha }$-sense to a $C^{1,\alpha }$-continuous Riemannian manifold which is isometric to the original metric space. Moreover, we prove that the flow is uniquely determined by the initial condition, up to isometry.

We show that smooth polarized Fano fibrations have no nontrivial finite covers. Using results by Sun-Zhang and Wylie, it follows that shrinking Kähler-Ricci solitons are simply-connected.

We study Perelman's W-entropy functional on finite-dimensional RCD spaces, a synthetic generalization of spaces with Bakry-Émery Ricci curvature bounded from below. We rigorously justify the formula for the time derivative of the W-entropy and derive its monotonicity and rigidity properties. Additionally, we establish bounds for solutions of the heat equation, which are of independent interest.

We show that any non-collapsed finite time singularity of the Ricci flow on a compact Kähler surface is of Type I. Combined with a previous result of the first author, Cifarelli, and Deruelle, it follows that any such singularity is modeled on the shrinking Ricci soliton of Feldman-Ilmanen-Knopf on the total space of the line bundle ${\mathcal{O}}_{{\mathbb{P}}^1}(-1)\to {\mathbb{P}}^1$.

In this paper, we study several properties of Sasaki-Ricci solitons as singularity models of the Sasaki-Ricci flow. First, we establish several fundamental equations for Sasaki-Ricci solitons, which enable us to derive potential estimates and prove the positivity of the scalar curvature. Then we present two criteria for the transverse rigidity of Sasaki-Ricci solitons. As essential applications, we prove that any low-dimensional Sasaki-Ricci soliton with constant scalar curvature must be Sasaki-Einstein, and that any Sasaki-Ricci soliton with harmonic Weyl tensor is a finite quotient of the sphere.

In this article, we prove an $\epsilon$-regularity theorem for Perelman's reduced volume. We show that on a Ricci flow, if Perelman's reduced volume is close to $1$, then the curvature radius at the base point cannot be too small.

We show that, up to biholomorphism, a given noncompact complex manifold only admits one shrinking gradient Kähler-Ricci soliton with Ricci curvature tending to zero at infinity. Our result does not require fixing the asymptotic data of the metric, nor fixing the soliton vector field. The method used to prove the uniqueness of the soliton vector field can be applied more widely, for example to show that conical Calabi-Yau metrics on a given complex manifold are unique up to biholomorphism. We also use it to prove that if two polarized Fano fibrations, as introduced by Sun-Zhang, are biholomorphic, then they are isomorphic as algebraic varieties.

We show that the system of vacuum Einstein equations (i.e., Ricci-flat metrics) with two hypersurface-orthogonal, commuting Killing vector fields in $d\ge 5$ dimensions is invariant under the action of a one-parameter Lie group, and the group action on any metric can be expressed in a closed, universal form. This enables the generation of a one-parameter family of solutions from any given "seed" solution of the system without solving additional equations, as well as one-parameter families of local steady Ricci solitons. This extends the Lie point symmetry in four dimensions, found earlier for axisymmetric static vacuum systems, and provides the first example of solution generation in higher-dimensional vacuum Einstein equations that can be realized purely algebraically.

We show that a general class of singular Kähler metrics with Ricci curvature bounded below define Kähler currents. In particular the result applies to singular Kähler-Einstein metrics on klt pairs, and an analogous result holds for Kähler-Ricci solitons. In addition we show that if a singular Kähler-Einstein metric can be approximated by smooth metrics on a resolution whose Ricci curvature has negative part that is bounded uniformly in $L^p$ for $p>\frac{2n-1}{n}$, then the metric defines an RCD space.

We demonstrate that any four-dimensional shrinking Ricci soliton $(\mathcal{B}\times {\mathbb{S}}^2,g)$, where $\mathcal{B}$ is any two-dimensional complete noncompact surface and $g$ is a warped product metric over the base $\mathcal{B}$, has to be isometric to the generalized cylinder ${\mathbb{R}}^2\times {\mathbb{S}}^2$ equipped with the standard cylindrical metric. After completing this classification, we study Ricci flow solutions that are multiply warped products – but not products – and provide rigorous examples of the formation of generalized cylinder singularity models ${\mathbb{R}}^k\times {\mathbb{S}}^{\ell }$.

Thurston's triangulation conjecture asserts that every hyperbolic 3-manifold admits a geometric decomposition into ideal hyperbolic tetrahedra, a result proven only for certain special 3-manifolds. This paper presents combinatorial Ricci flow as a systematic and general approach to addressing Thurston's triangulation conjecture, showing that the flow converges if and only if the triangulation is geometric. First, we prove the rigidity of the most general hyperbolic polyhedral 3-manifolds constructed by isometrically gluing partially truncated and decorated hyperbolic tetrahedra, demonstrating that the metrics are uniquely determined by cone angles modulo isometry and decoration changes. Then, we demonstrate that combinatorial Ricci flow evolves polyhedral metrics toward complete hyperbolic structures with geometric decompositions when convergent. Conversely, the existence of a geometric triangulation guarantees flow convergence.

A ${\mathrm{G}}_2$-structure on a $7$-manifold $M$ is called a ${\mathrm{G}}_{2}T$-structure if $M$ admits a ${\mathrm{G}}_2$-connection ${\nabla }^T$ with totally skew-symmetric torsion $T_{\phi }$. If furthermore, $T_{\phi }$ is closed then it is called a strong ${\mathrm{G}}_{2}T$-structure. In this paper we investigate the geometry of (strong) ${\mathrm{G}}_{2}T$-manifolds in relation to its curvature, $S^1$ action and almost Hermitian structures. In particular, we study the Ricci flatness condition of ${\nabla }^T$ and give an equivalent characterisation in terms of geometric properties of the ${\mathrm{G}}_2$ Lee form. Analogous results are also obtained for almost Hermitian $6$-manifolds with skew-symmetric Nijenhuis tensor. Moreover, by considering the $S^1$ reduction by the dual of the ${\mathrm{G}}_2$ Lee form, we show that Ricci-flat strong ${\mathrm{G}}_{2}T$-structures correspond to solutions of the $\mathrm{SU}(3)$ heterotic system on certain almost Hermitian half-flat $6$-manifolds. Many explicit examples are described and in particular, we construct the first examples of strong ${\mathrm{G}}_{2}T$-structures with ${\nabla }^T$ not Ricci flat. Lastly, we classify ${\mathrm{G}}_2$-flows inducing gauge fixed solutions to the generalised Ricci flow akin to the pluriclosed flow in complex geometry. The approach is this paper is based on the representation theoretic methods due to Bryant.

The notion of Courant algebroid relation is used to introduce a definition of relation between divergence operators on Courant algebroids. By introducing invariant divergence operators, a notion of generalised T-duality between divergences is presented through an existence and uniqueness result for related divergence operators on T-dual pairs of exact Courant algebroids, which naturally incorporates the dilaton shift. When combined with the notion of generalised isometry, this establishes circumstances under which generalised Ricci tensors are related, proving that T-duality is compatible with generalised string background equations. This enables an analysis of the compatibility between T-duality and generalised Ricci flow, showing that the T-dual of a solution of generalised Ricci flow is also a solution of generalised Ricci flow. Our constructions are illustrated through many explicit examples.

Let $(\overline{M},g_0)$ be a $2$-D compact surface with boundary $\partial M$ and its interior $M$. We show that for a large class of initial and boundary data, the initial-boundary value problem of the normalized Ricci flow $(1.10)-(1.12)$, with prescribed geodesic curvature $\psi$ on $\partial M$, has a unique solution for all $t>0$, and it converges to the complete hyperbolic metric locally uniformly in $M$. Here the natural condition that $\psi >0$ causes the main difficulty in the a priori estimates in the corresponding initial-boundary problem $(1.15)-(1.17)$ of the parabolic equations, for which an auxiliary Cauchy-Dirichlet problem is introduced. We also provide examples of the boundary data $\psi$ which fits well with the natural asymptotic behavior of the geodesic curvature, but the solution to $(1.10)-(1.12)$ fails to converge to the complete hyperbolic metric.

In his groundbreaking work from 2002, Perelman introduced two fundamental monotonic quantities: the reduced volume and the entropy. While the reduced volume was motivated by the Bishop-Gromov volume comparison applied to a suitably constructed $N$-space, which becomes Ricci-flat as $N\to \infty$, Perelman did not provide a corresponding explanation for the origin of the entropy. In this article, we demonstrate that Perelman's entropy emerges as the limit of Colding's monotonic volume for harmonic functions on Ricci-flat manifolds, when appropriately applied to Perelman's $N$-space.

We give biLipschitz models for the Ricci flow on some 4-manifolds (minimal surfaces of general type), exhibiting a combination of expanding and static behavior.

This paper is concerned with the geometry of principal orbits in quaternionic Kähler manifolds $M$ of cohomogeneity one. We focus on the complete cohomogeneity one examples obtained from the non-compact quaternionic Kähler symmetric spaces associated with the simple Lie groups of type A by the one-loop deformation. We prove that for zero deformation parameter the principal orbits form a fibration by solvsolitons (nilsolitons if $4n=\dim M=4$). The underlying solvable group is non-unimodular if $n>1$ and is the Heisenberg group if $n=1$. We show that under the deformation, the hypersurfaces remain solvmanifolds but cease to be Ricci solitons.

The full classification of Riemannian $3$-symmetric spaces is presented. Up to Riemannian products the main building blocks consist in (possibly symmetric) spaces with semisimple isometry group, nilpotent Lie groups of step at most $2$ and spaces of type III and IV. For the most interesting family of examples, the Type III spaces, we produce an explicit description including results concerning the moduli space of all $3$-symmetric metrics living on a given Type III space. Each moduli space contains a unique distinguished point corresponding to an (almost-Kähler) expanding Ricci soliton metric. For certain classes of 3-symmetric metrics there are many different groups acting transitively and isometrically on a fixed Riemannian 3-symmetric space. The construction of expanding Ricci solitons on spaces of Type III is also shown to generalize to any effective representation of a simple Lie group of non-compact type, yielding a very general construction of homogeneous Ricci solitons. We also give a procedure to compute the isometry group of any Ambrose–Singer space.

We develop different synthetic notions of Ricci flow in the setting of time-dependent metric measure spaces based on ideas from optimal transport. They are formulated in terms of dynamic convexity and local concavity of the entropy along Wasserstein geodesics on the one hand and in terms of global and short-time asymptotic transport cost estimates for the heat flow on the other hand. We show that these properties characterise smooth (weighted) Ricci flows. Further, we investigate the relation between the different notions in the non-smooth setting of time-dependent metric measure spaces.

We establish a rigidity theorem for Brendle and Hung's recent systolic inequality, which involves Gromov's notion of $T^{⋊}$-stabilized scalar curvature. Our primary technique is the construction of foliations by free boundary weighted constant mean curvature hypersurfaces, enabling us to generalize several classical scalar curvature rigidity results to the $T^{⋊}$-stabilized setting. Additionally, we develop a monotone quantity using Ricci flow coupled with a heat equation, which is essential for rigidity analysis.

We construct and classify all polynomial growth solutions to certain drift-harmonic equations on complete manifolds with paraboloidal asymptotics. These encompass the natural drift-harmonic equations on certain steady gradient Ricci solitons. Specifically, we show that all drift-harmonic functions with polynomial growth asymptotically separate variables, and compute the dimensions of spaces of drift-harmonic functions with a given polynomial growth rate. The proof uses an inductive argument that alternates between constructing and asymptotically controlling drift-harmonic functions.

In this paper, we extend the work of Ge-Hua-Zhou on combinatorial Ricci flows for ideal circle patterns to combinatorial Calabi flows in both hyperbolic and Euclidean background geometry. We prove the solution to the combinatorial Calabi flows with any given initial Euclidean (hyperbolic resp.)ideal circle pattern exists for all time and converges exponentially fast to a flat cone metric (hyperbolic resp.) on a given surface.

The article examines the impact of 16 key parameters of the Georgian economy on economic inequality, using the Perelman model and Ricci flow mathematical methods. The study aims to conduct a deep analysis of the impact of socio-economic challenges and technological progress on the dynamics of the Gini coefficient. The article examines the following parameters: income distribution, productivity (GDP per hour), unemployment rate, investment rate, inflation rate, migration (net negative), education level, social mobility, trade infrastructure, capital flows, innovative activities, access to healthcare, fiscal policy (budget deficit), international trade (turnover relative to GDP), social protection programs, and technological access. The results of the study confirm that technological innovations and social protection programs have a positive impact on reducing inequality. Productivity growth, improving the quality of education, and strengthening R&D investments increase the possibility of inclusive development. Sensitivity analysis shows that social mobility and infrastructure are important factors that affect economic stability. The accuracy of the model is confirmed by high R^2 values (80-90%) and the statistical reliability of the Z-statistic (<0.05). The study uses Ricci flow methods, which allow for a geometric analysis of the transformation of economic parameters in time and space. Recommendations include the strategic introduction of technological progress, the expansion of social protection programs, improving the quality of education, and encouraging international trade, which will contribute to economic sustainability and reduce inequality. The article highlights multifaceted approaches that combine technological innovation and responses to socio-economic challenges to ensure sustainable and inclusive economic development.