Papers from 2024

We give an application of a Huisken monotonicity-type formula for the mean curvature flow in a compact smooth manifold with a Riemannian metric that evolves by a shrinking self-similar solution of the extended Ricci flow. Our investigation builds on previous articles by Huisken and the third author, as we apply their techniques to establish new results in this geometric setting. Moreover, under some natural geometric assumptions, the noncompact case is also solved

This manuscript investigates the curvature and topological properties of certain $\infty$-Einstein Finsler metrics on Finsler metric measure spaces. By imposing symmetry conditions, we construct a series of special metrics and analyze their equivalence on special manifolds. Provided a Ricci curvature bound, we establish a linear growth lower bound estimate for the S-curvature and the distortion, revealing the interplay between curvature and measure on $\infty$-Einstein Finsler manifolds. Furthermore, by introducing scalar curvature and imposing a linear growth lower bound condition, we derive upper and lower bounds for the distortion, S-curvature, and the scalar curvature itself on asymmetric essential gradient Ricci solitons with certain non-Riemannian curvature constraints. These results yield direct topological finiteness conclusions for some forward-complete $\infty$-Einstein Finsler manifolds. Our work partially addresses Gromov's conjecture of scalar curvature in the context of Finsler metric measure spaces and provides a foundation for further research in geometric analysis within general Finsler geometry.

We construct a continuous 3-parameter family of non-shrinking Ricci solitons complex line bundles $O(k)$ over ${\mathbb{CP}}^{2m+1}$, where the base space is not necessarily Kähler–Einstein. Each $O(k)$ with $k\in [3,2m+1]$ admits at least one asymptotically conical (AC) Ricci-flat metric in this family. For each $O(k)$ with $k\ge 3$, the family includes infinitely many asymptotically paraboloidal (AP) steady Ricci soliton.

Recent interest among geometers in $f$-structures of K. Yano is due to the study of topology and dynamics of contact foliations, which generalize the flow of the Reeb vector field on contact manifolds to higher dimensions. Weak metric structures introduced by V. Rovenski and R. Wolak as a generalization of Hermitian and Kähler structures, as well as $f$-structures, allow a fresh look at the classical theory. In this paper, we study a new $f$-structure of this kind, called the weak $\beta$-Kenmotsu $f$-structure, as a generalization of K. Kenmotsu's concept. We prove that a weak $\beta$-Kenmotsu $f$-manifold is locally a twisted product of the Euclidean space and a weak Kähler manifold. Our main results show that such manifolds with $\beta =const$ and equipped with an $\eta$-Ricci soliton structure whose potential vector field satisfies certain conditions are $\eta$-Einstein manifolds of constant scalar curvature.

We would like to study new Ricci flow invariant curvature conditions. Specifically, we provide quantitative evidence for an unpublished conjecture of Böhm and Wilking. As an application, we study the topology of manifolds with pinched curvature.

The Swampland Distance Conjecture postulates the emergence of an infinite tower of massless states when approaching infinite-distance points in moduli space. However, most string backgrounds are supported by fluxes, and therefore depart from the purely geometric paradigm. This fact requires an extension of the Swampland conjectures to scalar field spaces with non-trivial potentials, rather than just moduli spaces. To address this task, we utilise geometric flows, in particular generalised Ricci flow, to probe the associated scalar field spaces. Considering internal spaces supported by three-form fluxes, we first show that the distance defined in terms of the Perelman entropy functional needs to be refined in order to encompass fluxes. Doing so, we extend the Ricci Flow Conjecture to include Kalb-Ramond flux besides the metric and the dilaton field. This allows us to probe infinite-distance points within these scalar field spaces in a purely geometric way. We subsequently construct a geometric flow for internal manifolds supported by Ramond-Ramond fluxes and discuss its role in the Ricci Flow Conjecture. Our analysis suggests that in the presence of fluxes the Distance Conjecture might be better characterised in terms of a cost function on the space of metrics, rather than a genuine distance.

In this paper, we study the fundamental group of the complete steady gradient Ricci soliton with nonnegative sectional curvature. We prove that the fundamental group of such a Ricci soliton is either trivial or infinite. As a corollary, we show that an $n$-dimensional complete $\kappa$-noncollapsed steady gradient Ricci soliton with nonnegative sectional curvature must be diffeomorphic to ${\mathbb{R}}^n$.

In this paper, we study Kähler-Ricci solitons on bounded pseudoconvex domains in ${\mathbb{C}}^n$ with $C^2$ boundary. Under suitable assumptions, we prove that such solitons must be Kähler-Einstein. Building on Huang and Xiao's resolution of Cheng's conjecture, we further establish an analogous result for Bergman Kähler-Ricci solitons. Several model domains are presented to illustrate our results.

We proved that on every Stiefel manifold $V_2{\mathbb{R}}^n\cong \mathrm{SO}(n)/\mathrm{SO}(n-2)$ with $n\ge 3$ the normalized Ricci flow preserves the positivity of the Ricci curvature of invariant Riemannian metrics with positive Ricci curvature. Moreover, the normalized Ricci flow evolves all metrics with mixed Ricci curvature into metrics with positive Ricci curvature in finite time. From the point of view of the theory of dynamical systems we proved that for every invariant set $\Sigma$ of the normalized Ricci flow on $V_2{\mathbb{R}}^n$ defined as $x_1^{n-2}{x}_2^{n-2}{x}_3=c$, $c>0$, there exists a smaller invariant set $\Sigma \cap {\mathcal{R}}_{+}$ for every $n\ge 3$, where ${\mathcal{R}}_{+}$ is the domain in ${\mathbb{R}}_{+}^3$ responsible for parameters $x_1,x_2,x_3>0$ of invariant Riemannian metrics on $V_2{\mathbb{R}}^n$ admitting positive Ricci curvature.

We show that if $X$ is a smooth Fano manifold which caries a Kähler Ricci soliton, then the canonical cone of the product of $X$ with a complex projective space of sufficiently large dimension is a Calabi–Yau cone. This can be seen as an asymptotic version of a conjecture by Mabuchi and Nikagawa. This result is obtained by the openness of the set of weight functions $v$ over the momentum polytope of a given smooth Fano manifold, for which a $v$-soliton exists. We discuss other ramifications of this approach, including a Licherowicz type obstruction to the existence of a Kähler Ricci soliton and a Fujita type volume bound for the existence of a $v$-soliton.

This paper explores the behavior of the torsional rigidity of a precompact domain as the ambient manifold evolves under a geometric flow. Specifically, we derive bounds on torsional rigidity under the Ricci Flow for Heisenberg spaces and homogeneous spheres. Additionally, we establish bounds under the Inverse Mean Curvature Flow for strictly convex, free-boundary, disk-type hypersurfaces within a ball. In this latter case, by extending the analysis to the maximal existence time of the flow, we obtain inequalities of comparison with the flat disk for both volume and torsional rigidity.

In the present paper we study $\eta$-Ricci solitons on Kenmotsu 3-manifolds. Moreover, we consider $\eta$-Ricci solitons on Kenmotsu 3-manifolds with Codazzi type of Ricci tensor and cyclic parallel Ricci tensor. Beside these, we study $\phi$-Ricci symmetric $\eta$-Ricci soliton on Kenmotsu 3-manifolds. Also Kenmotsu 3-manifolds satisfying the curvature condition $R.R=Q(S,R)$ is considered. Finally, an example is constructed to prove the existence of a proper $\eta$-Ricci soliton on a Kenmotsu 3-manifold.

In this paper we survey some results on Ricci flowing non-smooth initial data. Among other things, we give a non-exhaustive list of various weak initial data which can be evolved with the Ricci flow. We also survey results which show that various curvature lower bounds will, possibly up to a constant, be preserved, if we start with such possibly non-smooth initial data. Some proofs/proof sketches are given in certain cases. A list of some open problems related to these areas is given in the last section of the paper.

We prove that there is no nontrivial $L^2$-integrable harmonic 1-form on noncompact complete gradient steady Ricci solitons or noncompact complete gradient shrinking Kähler-Ricci solitons. As an application, it can be used to distinguish certain flat vector bundles that arise from fundamental group representations into $SL(r,\mathbb{C})$.

Let $(M,g,f)$ be a $5$-dimensional complete noncompact gradient shrinking Ricci soliton with the equation $Ric+{\nabla }^{2}f=\lambda g$, where $\text{Ric}$ is the Ricci tensor and ${\nabla }^{2}f$ is the Hessian of the potential function $f$. We prove that it is a finite quotient of ${\mathbb{R}}^2\times {\mathbb{S}}^3$ if $M$ has constant scalar curvature $R=3\lambda$.

We show how solutions to the Ricci flow on Lorentzian manifolds, along with its generalizations, can be linked to Einstein's field equations. The approach involves deformations of the matter sector that are generated by quadratic functionals of the stress-energy tensor. We provide illustrative examples by explicitly constructing analytical solutions within maximally symmetric spacetimes and in the context of Born-Infeld's nonlinear electrodynamics. Finally, we discuss configurations involving global topological monopoles, emphasizing the versatility of this approach across various geometric and physical settings.

A complex Monge-Ampère equation for differential $(p,p)$-forms is introduced on compact Kähler manifolds. For any $1\le p<n$, we show the existence of smooth solutions unique up to adding constants. For $p=1$, this corresponds to the Calabi-Yau theorem proved by S. T. Yau, and for $p=n-1$, this gives the Monge-Ampère equation for $(n-1)$ plurisubharmonic functions studied by Tosatti-Weinkove. For other $p$ values, this defines a non-linear PDE that falls outside of the general framework of Caffarelli-Nirenberg-Spruck. Further, we define a geometric flow for higher-order forms that preserves their cohomology classes, and extends the Kähler-Ricci flow naturally to $(p,p)$-forms. As a consequence of our main theorem, we show that this flow exists in a maximal time interval and can be shown to converge under some assumptions. A modified flow is introduced and the convergence of the associated normalized flow is shown.

Suppose $(M^n,g,f)$ is a complete shrinking gradient Ricci soliton. We give several rigidity results under some natural conditions, generalizing the results in. Using maximum principle, we prove that shrinking gradient Ricci soliton with constant scalar curvature $R=1$ is isometric to a finite quotient of ${\mathbb{R}}^2\times {\mathbb{S}}^2$, giving a new proof of the main results of Cheng-Zhou.

On a connected finite graph, we propose an evolution of weights including Ollivier's Ricci flow as a special case. During the evolution process, on each edge, the speed of change of weight is exactly the difference between the Wasserstein distance related to two probability measures and certain graph distance. Here the probability measure may be chosen as an $\alpha$-lazy one-step random walk, an $\alpha$-lazy two-step random walk, or a general probability measure. Based on the ODE theory, we show that the initial value problem has a unique global solution. A discrete version of the above evolution is applied to the problem of community detection. Our algorithm is based on such a discrete evolution, where probability measures are chosen as $\alpha$-lazy one-step random walk and $\alpha$-lazy two-step random walk respectively. Note that the latter measure has not been used in previous works. Moreover, only one surgery needs to be performed after the last iteration, which makes our algorithm much simpler than earlier ones based on Lin-Lu-Yau's Ricci curvature.

In this article we study generalizations of the inhomogeneous Burgers equation. First at the operator level, in the sense that we replace classical differential derivations by operators with certain properties, and then we increase the spatial dimensions of the Burgers equation, which is usually studied in one spatial dimension. This allows us, in one dimension, to find mathematical relationships between solutions of hyperbolic Brownian motion and the Burgers equations, which usually study the behaviour of mechanical fluids, and also, through appropriate transformations, to obtain in some cases exact solutions that depend on Hermite polynomials composed of appropriate functions. In the multi-dimensional case, this generalization allows us, by means of the method of invariant spaces, to find exact solutions on Riemannian and pseudo-Riemannian varieties, such as Schwarzschild and Ricci Solitons space, with time dictated by fractional derivatives, such as a Caputo-type operator of fractional evolution.

We first determine the asymptotic cone of the steady gradient Kähler-Ricci soliton of the Taub-NUT type constructed by Apostolov and Cifarell. Then we study a special case and prove that it is an ALF Calabi-Yau metric in a certain sense. Finally we construct new ALF Calabi-Yau metrics on crepant resolution of its quotients modeled on it using the method of Tian-Yau-Hein.

This paper proposes a theoretical framework for modeling and optimizing the bounded functions based on the Fourier series approximation and Ricci flow. Specifically, the initial manifold, ${\mathcal{M}}_0$ is approximated using Fourier series approximation in conjunction with the center and boundary sampling procedure introduced in the paper. The manifold is iteratively evolved using an algorithm that involves sampling along geodesic hyper-sphere defined by the Riemannian metric tensor. Thus obtained surrogate manifold is optimized by applying inverse Ricci flow i.e. instead of regularizing the manifold, flow allows for the high curvature regions to blow into finite time singularities. This allows for the singularities to occur at potential global optima assuming the deviation of the manifold at any point is smaller than the optimum. In addition, the error bound is established on the accuracy of the surrogate manifold. Finally, the proposed method is tested on stochastic sampling from five benchmark functions to illustrate the utility of this method.

This paper explores the evolution and monotonicity of geometric constants within the framework of extended Ricci flows, incorporating variable coupling parameters. Building on Hamiltons foundational Ricci flow and subsequent extensions by List (2008), we introduce modifications to the extended Ricci flow by varying parameters that affect the interaction between the metric and scalar fields. Specifically, we modify the coefficients in the evolution equations governing geometric constants, thereby introducing new degrees of freedom in the analysis. The primary contributions include deriving evolution formulas for the modified geometric constant lambda under the extended and normalized extended Ricci flows, and proving conditions under which monotonicity is maintained.

In this paper, we investigate the evolution of certain functionals involving higher powers of a scalar quantity $F$ under Bernard List's extended Ricci flow on a compact Riemannian manifold. By deriving explicit expressions for the time derivative of integrals of the form ${\int }_M{F}^n\cdot \frac{\partial F}{\partial t}d\mu$ for various powers $n$, we explore the intricate interplay between geometric quantities and scalar functions without making any assumptions about the manifold, the scalar field $\Phi$, or the function $u$.

We establish the existence of two 3-parameter families of non-Einstein, non-shrinking Ricci solitons: one on ${\mathbb{H}}^{m+1}$ and one on ${\mathbb{HP}}^{m+1}\backslash \{\ast \}$. Each family includes a continuous 1-parameter subfamily of asymptotically paraboloidal (non-collapsed) steady Ricci solitons, with the Jensen sphere as the base. Additionally, we extend this result by proving the existence of a 2-parameter family on ${\mathbb{O}}^2$, which contains a 1-parameter subfamily of asymptotically paraboloidal steady Ricci solitons based on the Bourguignon–Karcher sphere.

Let $B$ be a Kähler-Einstein Fano manifold, and $L\to B$ be a suitable root of the canonical bundle. We give a construction of complete Calabi-Yau metrics and gradient shrinking, steady, and expanding Kähler-Ricci solitons on the total space $M$, ${\dim }_{\mathbb{C}}M=n$ of certain vector bundles $E\to B$, composed of direct sums of powers of $L$. We employ the theory of hamiltonian 2-forms [2, 3] as an Ansatz, thus generalizing recent work of the author and Apostolov on ${\mathbb{C}}^n$ [5], as well as that of Cao, Koiso, Feldman-Ilmanen-Knopf, Futaki-Wang, and Chi Li [10, 26, 23, 24, 30] when $E$ has Calabi symmetry. As a result, we obtain new examples of asymptotically conical Kähler shrinkers, Calabi-Yau metrics with ALF-like volume growth, and steady solitons with volume growth $R^{\frac{4n-2}{3}}$.

We examine a non-axisymmetric perturbation of a family of axisymmetric toric Einstein manifolds and Ricci solitons studied in Firester-Tsiamis (2024). We establish a rigidity result stating that these axisymmetric Ricci solitons do not admit constant-angle non-axisymmetric perturbations except for conformally flat cases. For these new cases, our result leads to an explicit description of the Einstein metrics and a classification of the Ricci solitons under a volume-collapsing ansatz.

In stark contrast to lower dimensions, we produce a plethora of ancient and immortal Ricci flows in real dimension $4$ with Einstein orbifolds as tangent flows at infinity. For instance, for any $k\in {\mathbb{N}}_0$, we obtain continuous families of non-isometric ancient Ricci flows on $\#k({\mathbb{S}}^2\times {\mathbb{S}}^2)$ depending on a number of parameters growing linearly in $k$, and a family of half-PIC ancient Ricci flows on ${\mathbb{CP}}^2\#{\mathbb{CP}}^2$. The ancient/immortal dichotomy is determined by a notion of linear stability of orbifold singularities with respect to the expected way for them to appear along Ricci flow: by bubbling off Ricci-flat ALE metrics. We discuss the case of Ricci solitons orbifolds and motivate a conjecture that spherical and cylindrical solitons with orbifold singularities, which are unstable in our sense, should not appear along Ricci flow by bubbling off Ricci-flat ALE metrics.

We use Lott's functional and construct a new functional to derive rigidity results for invariant Ricci flow blowdown limits on nilpotent principal bundles with zero associated curvature. Consequently, we prove that the blowdown limit is locally an expanding Ricci soliton when the structure group is the three dimensional Heisenberg group. In addition, we classify this soliton when the base manifold is one dimensional. This, together with Lott's work in the abelian setting, yields a complete local classification of invariant Ricci flow blowdown limits on four dimensional, nilpotent principal bundles.

In this paper, we build connections between Kähler-Ricci shrinkers, i.e., complete (possibly non-compact) shrinking gradient Kähler-Ricci solitons, and algebraic geometry. In particular, we (1). prove that a Kähler-Ricci shrinker is naturally a quasi-projective variety, using birational algebraic geometry; (2). formulate a conjecture relating the existence of Kähler-Ricci shrinkers and K-stability of polarized Fano fibrations, which unifies and extends the YTD type conjectures for Kähler-Einstein metrics, Ricci-flat Kähler cone metrics and compact Kähler-Ricci shrinkers; (3). formulate conjectures connecting tangent flows at singularities of Kähler-Ricci flows and algebraic geometry, via a 2-step degeneration for the weighted volume of a Fano fibration.

In this paper we study $n$-dimensional Ricci flows $(M^n,g(t){)}_{t\in [0,T)},$ where $T<\infty$ is a potentially singular time, and for which the spatial $L^p$ norm, $p>\frac{n}{2}$, of the scalar curvature is uniformly bounded on $[0,T).$ In the case that $M$ is closed and four dimensional, we explain why non-collapsing estimates hold and how they can be combined with integral bounds on the Ricci and full curvature tensor of the prequel paper of the authors, as well as non-inflating estimates (already known due to works of Bamler), to obtain an improved space time integral bound of the Ricci curvature. As an application of these estimates, we show that if we further restrict to $n=4$, then the solution convergences to an orbifold as $t\to T$ and that the flow can be extended using the Orbifold Ricci flow to the time interval $[0,T+\sigma )$ for some $\sigma >0.$ We also prove local versions of many of the results mentioned above.

We elaborate on nonmetric geometric flow theory and metric-affine gravity with applications in modern cosmology. Two main motivations for our research follow from the facts that 1) cosmological models for $f(Q)$ modified gravity theories, MGTs, are efficient for describing recent observational data provided by the James Webb Space Telescope; and 2) the statistical thermodynamic properties of such nonmetric locally anisotropic cosmological models can be studied using generalizations of the concept of G. Perelman entropy. We derive nonmetric distorted R. Hamilton and Ricci soliton equations in such canonical nonholonomic variables when corresponding systems of nonlinear PDEs can be decoupled and integrated in general off-diagonal forms. This is possible if we develop and apply the anholonomic frame and connection deformation method involving corresponding types of generating functions and generating sources encoding nonmetric distortions. Using such generic off-diagonal solutions (when the coefficients of metrics and connections may depend generically on all spacetime coordinates), we model accelerating cosmological scenarios with quasi-periodic gravitational and (effective) matter fields; and study topological and nonlinear geometric properties of respective dark energy and dark matter, DE and DM, models. As explicit examples, we analyze some classes of nonlinear symmetries defining topological quasicrystal, QC, phases which can modified to generate other types of quasi-periodic and locally anisotropic structures. The conditions when such nonlinear systems possess a behaviour which is similar to that of the Lambda cold dark matter ($\Lambda$CDM) scenario are stated. We conclude that nonmetric geometric and cosmological flows can be considered as an alternative to the $\Lambda$CDM concordance models and speculate on how such theories can be elaborated.

We elaborate on a model of nonassociative and noncommutative gauge gravity for the de Sitter gauge group $SO(4,1)$ embedding extensions of the affine structure group $Af(4,1)$ and the Poincaré group $ISO(3,1)$. In string theory, such nonassociative gauge gravity theories are determined by star product R-flux deformations. They are new avenues to quantum gravity and geometric and quantum information theories. We analyze physically important and geometric thermodynamic properties of new classes of generic off-diagonal cosmological solitonic solutions encoding nonassociative effective sources. Particularly, we focus on modelling by such solutions of locally anisotropic and inhomogeneous dark matter and dark energy structures generated as nonassociative solitonic hierarchies. Such accelerating cosmological evolution scenarios can't be described in the framework of the Bekenstein-Hawking thermodynamic formalism. This motivates a change in the gravitational thermodynamic paradigm by considering nonassociative and relativistic generalizations of the concept of W-entropy in the theory of Ricci flows. Finally, we compute the corresponding modified G. Perelman's thermodynamic variables and analyze the temperature-like evolution of cosmological constants determined by nonassociative cosmological flows.

[Dedicated to Richard S. Hamilton on forty years of Ricci flow] Gradient Ricci solitons have garnered significant attention both as self-similar solutions and singularity models of the Ricci flow. This survey article starts with a list of examples; it also provides some geometric aspects of gradient Ricci solitons, including various asymptotic behaviors; finally, it discusses some recent results on classification and rigidity. In particular, this survey focuses on dimension four.

We consider the Kähler-Ricci flow $(X,\omega (t){)}_{t\in [0,T)}$ on a compact manifold where the time of singularity, $T$, is finite. We assume the existence of a holomorphic map from the Kähler manifold $X$ to some analytic variety $Y$ which admits a Kähler metric on a neighbourhood of the image of $X$ and that the pullback of this metric yields the limiting cohomology class along the flow. This is satisfied, for instance, by the assumption that the initial cohomology class is rational, i.e., $[{\omega }_0]\in H^{1,1}(X,\mathbb{Q})$. Under these assumptions we prove an $L^4$-like estimate on the behaviour of the Ricci curvature and that the Riemannian curvature is Type $I$ in the $L^2$-sense.

This paper explores a novel approach to modeling the positional dynamics of stars using discrete dynamical systems. We define star evolution through discrete-time update rules based on right ascension, declination, and distance, incorporating chaotic behavior via nonlinear functions and external perturbations. By applying Ricci flow and Riemannian metrics, we provide new insights into the positional dynamics of stars. Theoretical computations of Perelman entropy are used to assess system complexity, with high-precision Runge-Kutta methods ensuring accurate solutions for our chaotic model. We quantify chaos using Lyapunov exponents and perform bifurcation analysis to study how parameter variations affect the dynamics. Comparing our model to the Lorenz attractor reveals both similarities and unique characteristics in stellar dynamics. Our results show that entropy increases exponentially, indicating that predicting star positions with precision becomes increasingly challenging over time. This study advances the understanding of chaos in celestial systems and contributes to dynamical systems theory by integrating chaos theory with astronomical modeling.

We proved that the normalized Ricci flow does not preserve the positivity of Ricci curvature of Riemannian metrics on every generalized Wallach space with $a_1+a_2+a_3\le 1/2$, in particular on the spaces $\mathrm{SU}(k+l+m)/\mathrm{SU}(k)\times \mathrm{SU}(l)\times \mathrm{SU}(m)$ and $\mathrm{Sp}(k+l+m)/\mathrm{Sp}(k)\times \mathrm{Sp}(l)\times \mathrm{Sp}(m)$ independently on $k,l$ and $m$. The positivity of Ricci curvature is preserved for all original metrics with $\mathrm{Ric}>0$ on generalized Wallach spaces $a_1+a_2+a_3>1/2$ if the conditions $4{\left(a_j+a_k\right)}^2\ge (1-2a_i)(1+2a_i{)}^{-1}$ hold for all $\{i,j,k\}=\{1,2,3\}$. We also established that the spaces $\mathrm{SO}(k+l+m)/\mathrm{SO}(k)\times \mathrm{SO}(l)\times \mathrm{SO}(m)$ satisfy the above conditions for $\max \{k,l,m\}\le 11$, moreover, additional conditions were found to keep $\mathrm{Ric}>0$ in cases when $\max \{k,l,m\}\le 11$ is violated. Similar questions have also been studied for all other generalized Wallach spaces given in the classification of Yuri\uı Nikonorov.

We provide new type of decay estimate for scalar curvatures of steady gradient Ricci solitons. We also give certain upper bound for the diameter of a Riemannian manifold whose $\infty$-Bakry–Emery Ricci tensor is bounded by some positive constant from below. For the proofs, we use $\mu$-bubbles introduced by Gromov.

We introduce new families of four-dimensional Ricci solitons of cohomogeneity two with volume collapsing ends. In a local presentation of the metric conformal to a product, we reduce the soliton equation to a degenerate Monge-Ampère equation for the conformal factor coupled with ODEs. We obtain explicit complete expanding solitons as well as abstract existence results for shrinking and steady solitons with boundary. These families of Ricci solitons specialize to classical examples of Einstein and soliton metrics. We also classify local solutions of this Monge-Ampère equation to prove rigidity for these solitons.

We extend some results known for the Kähler-Ricci flow to the Chern-Ricci flow regarding the independence of singularity types for long-time solutions. Specifically, we show that if a solution to the Chern-Ricci flow exists with uniformly bounded torsion and curvature, then any other solution starting from an initial metric of the same $\partial \bar{\partial }$ class will also exhibit uniform bounds on torsion and curvature.

We define a natural extension of pluriclosed flow aiming at constructing solutions of the Hull-Strominger system. We give several geometric formulations of this flow, which yield a series of a priori estimates for the flow and also for the Hull-Strominger system. The evolution equations are derived using the theory of string algebroids, a class of Courant algebroids which occur naturally in higher gauge theory. Using this, we interpret the flow as generalized Ricci flow and also as a higher/coupled version of Hermitian-Yang-Mills flow, proving furthermore that it is compatible with symmetry reduction. Regarding our main analytical results, we prove a priori $C^{\infty }$ estimates for uniformly parabolic solutions. This in particular settles the question of smooth regularity of uniformly elliptic solutions of the Hull-Strominger system, generalizing Yau's $C^3$ estimate for the complex Monge-Ampère equation. We prove global existence and convergence results for the flow on special backgrounds, and discuss a conjectural relationship of the flow to the geometrization of Reid's fantasy.

In recent work of Chan-Huang-Lee, it is shown that if a manifold enjoys uniform bounds on (a) the negative part of the scalar curvature, (b) the local entropy, and (c) volume ratios up to a fixed scale, then there exists a Ricci flow for some definite time with estimates on the solution assuming that the local curvature concentration is small enough initially (depending only on these a priori bounds). In this work, we show that the bound on scalar curvature assumption (a) is redundant. We also give some applications of this quantitative short-time existence, including a Ricci flow smoothing result for measure space limits, a Gromov-Hausdorff compactness result, and a topological and geometric rigidity result in the case that the a priori local bounds are strengthened to be global.

In this paper, we propose a modified Ricci flow, as well as a quasi-normalized Ricci flow, on arbitrary weighted graph. Each of these two flows has a unique global solution. In particular, these global existence and uniqueness results do not require an exit condition proposed by Bai et al in a recent work [2]. As applications, these two Ricci flows are applied to community detection for complex networks, including Karate Club, American football games, Facebook, as well as artificial networks. In our algorithms, unlike in [5,15], there is no need to perform surgery at every iteration, only one surgery needs to be performed after the last iteration. From three commonly used criteria for evaluating community detection algorithms, ARI, NMI and Q, we conclude that our algorithms outperform existing algorithms, including Ollivier's Ricci flow [5], normalized Ollivier's Ricci flow and normalized Lin-Lu-Yau's Ricci flow [15]. The codes for our algorithms are available at https://github.com/mjc191812/Modified-Ricci-Flow.

In this paper we prove that there exists a compact perturbation of the Ricci flat Taub-Bolt metric that evolves under the Ricci flow into a finite time singularity modelled on the shrinking solition FIK [5]. Moreover, this perturbation can be made arbitrarily L^2-small with respect to the Taub-Bolt metric. The method of proof closely follows the strategy adopted in [14], where the author constructs, via a Ważewski box argument, Ricci flows on compact manifolds which encounter finite singularities modelled on a given asymptotically conical shrinking soliton.

We introduce a new weighted version of the Hermite–Einstein equation, along with notions of weighted slope (semi/poly)stability, and prove that a vector bundle admits a weighted Hermite–Einstein metric if and only if it is weighted slope polystable. The new equation encompasses several well-known examples of canonical Hermitian metrics on vector bundles, including the usual Hermite–Einstein metrics, Kähler–Ricci solitons, and transversally Hermite–Einstein metrics on certain Sasaki manifolds. We prove that the equation arises naturally as a moment map, that solutions to the equation are unique up to scaling, and demonstrate a weighted Kobayashi–Lübke inequality satisfied by vector bundles admitting a weighted Hermite–Einstein metric. As an application of our techniques, we extend a bound of Tian on the Ricci curvature to a bound on a modified Ricci curvature, related to the existence of Kähler–Ricci solitons. Along the way, we introduce a new weighted vortex equation, as well as a weighted analogue of Gieseker stability. A key technical point is the application of a new extension of Inoue's equivariant intersection numbers to arbitrary weight functions on the moment polytope of a Kähler manifold with Hamiltonian torus action.

This paper reviews and extends the recently discovered connections between marginal and irrelevant stress-energy tensor deformations and gravity theories in arbitrary space-time dimensions. We start by discussing how $T\bar{T}$ and Root-$T\bar{T}$ deformations of two-dimensional field theories can be equivalently interpreted as the coupling of the undeformed matter sector to a gravity theory. We then extend this duality to higher-dimensional scenarios by using an approach that relies on the non-trivial eigenvalue degeneracy characterising the energy-momentum tensor of specific physical theories. We also explore incorporating dynamical degrees of freedom in the gravity sector, and show that the deformed space-time geometry induced by the $T\bar{T}$-like deformations defines a Ricci-Bourguignon flow of the metric tensor, which reduces to a Ricci flow in four dimensions. Finally, exploiting a dressing-type mechanism for the action functional characterizing a broad class of $T\bar{T}$-like deformations we study explicit examples, such as Einstein-Ricci solitons, $(d-1)$-form field theories, and spherically symmetric electrovacuum solutions.

In Riemannian geometry, Ricci soliton inequalities are an important field of study that provide profound insights into the geometric and analytic characteristics of Riemannian manifolds. An extensive study of Ricci soliton inequalities is given in this review article, which also summarizes their historical evolution, core ideas, important findings, and applications. We investigate the complex interactions between curvature conditions and geometric inequalities as well as the several kinds of Ricci solitons, such as expanding, steady, and shrinking solitons. We also go over current developments, unresolved issues, and possible paths for further study in this fascinating area.

This undergraduate thesis is focused on introducing the reader to concepts related to the search for topological obstructions to the existence of compact gradient shrinking Ricci soliton metrics in dimension four. It contains a discussion of the relevant background material for this subject. Furthermore, it introduces the problem of extending the Hitchin-Thorpe inequality to gradient shrinking Ricci soliton metrics and explores the limitations of current results in that direction. At last, the topic of compact Kaehler gradient shrinking Ricci solitons is introduced and the classification of these spaces is outlined in literature-study fashion.

In this short survey paper, we first recall the log gradient estimates for the heat equation on manifolds by Li-Yau, R. Hamilton and later by Perelman in conjunction with the Ricci flow. Then we will discuss some of their applications and extensions focusing on sharp constants and improved curvature conditions.

We classify spin ALE ancient Ricci flows and spin ALE expanding solitons with suitable groups at infinity. In particular, the only spin ancient Ricci flows with groups at infinity in $SU(2)$ and mild decay at infinity are hyperkähler ALE metrics. The main idea of the proof, of independent interest, consists in showing that the large-scale behavior of Perelman's $\mu$-functional on any ALE orbifold with non-negative scalar curvature is controlled by a renormalized ${\lambda }_{\mathrm{ALE}}$-functional related to a notion of weighted mass.

We study the flow of Hermitian metrics governed by the second Chern-Ricci form on a compact complex manifolds. The flow belongs to the family of Hermitian curvature flows introduced by Streets and Tian and it was considered by Lee in order to study compact Hermitian manifolds with almost negative Chern bisectional curvature. We show a regularity result on compact complex parallelizable manifolds and we prove that Chern-flat metrics are dynamically stable.

This paper investigates a kind of degenerated circle packings in hyperbolic background geometry. A main problem is whether a prescribed total geodesic curvature data can be realized by a degenerated circle packing or not. We fully characterize the sufficient and necessary conditions and show the uniqueness. Furthermore, we introduce the combinatoral Ricci flow to find the desired degenerated circle packed surface, analougus to the methods of Chow-Luo and Takatsu.

Any homogeneous expanding Ricci soliton is known to be isometric to a Lie subgroup of the solvable part of the Iwasawa decomposition associated with a symmetric space of non-compact type, with the metric induced as a submanifold. In this paper, we classify and analyze the geometry of such Lie subgroups with Ricci soliton induced metric when the symmetric spaces are complex hyperbolic spaces.

In this work, we study the short-time existence theory of Ricci-DeTurck flow starting from rough metrics which satisfy a Morrey-type integrability condition. Using the rough existence theory, we show the preservation and improvement of distributional scalar curvature lower bounds provided the singular set for such metrics is not too large. As an application, we use the Ricci flow smoothing to study the removable singularity for scalar curvature rigidity in the compact case under Morrey regularity conditions. Our result supplements those of Jiang-Sheng-Zhang.

We sharpen a gap theorem of Chan & Lee for nonnegative Ricci curvature manifolds that have positive asymptotic volume ratio and small enough scale-invariant integral curvature (so-called "curvature concentration"), by showing that the curvature concentration need only depend linearly on the asymptotic volume ratio. We prove the result by exhibiting a long-time Ricci flow solution with faster than $1/t$ curvature decay, which allows us to shift the limiting contradiction argument to time infinity and thus obtain an explicit bound on the size of the gap.

Using a size condition of the sharp log Sobolev functional (log entropy) near infinity only, we prove a rigidity result for ancient Ricci flows without sign condition on the curvatures. The result is also related to the problem of identifying type II ancient Ricci flows and their backward limits.

In this paper, we derive the uniform L^4-bound of the transverse conic Ricci curvature along the conic Sasaki-Ricci flow on a compact transverse log Fano Sasakian manifold M of dimension five and the space of leaves of the characteristic foliation is not well-formed. Then we first show that any solution of the conic Sasaki-Ricci flow converges in the Cheeger-Gromov sense to the unique singular orbifold conic Sasaki-Ricci soliton on M_infinite which is a S^1-orbibundle over the unique singular conic Keahler-Ricci soliton on a log del Pezzo orbifold surface. As a consequence, there exists a Keahler-Ricci soliton orbifold metric on its leave space which is a log del Pezzo orbifold surface. Second, we show that the conic Sasaki-Ricci soliton is the conic Sasaki-Einstein if M is transverse log K-polystable. In summary, we have the existence theorems of orbifold Sasaki-Ricci solitons and Sasaki-Einstein metrics on a compact quasi-regular Sasakian manifold of dimension five.

In this article, we characterize a Lorentzian manifold $\mathcal{M}$ with a semi-symmetric metric connection. At first, we consider a semi-symmetric metric connection whose curvature tensor vanishes and establish that if the associated vector field is a unit time-like torse-forming vector field, then $\mathcal{M}$ becomes a perfect fluid spacetime. Moreover, we prove that if $\mathcal{M}$ admits a semi-symmetric metric connection whose Ricci tensor is symmetric and torsion tensor is recurrent, then $\mathcal{M}$ represents a generalized Robertson-Walker spacetime. Also, we show that if the associated vector field of a semi-symmetric metric connection whose curvature tensor vanishes is a $f-$ Ric vector field, then the manifold is Einstein and if the associated vector field is a torqued vector field, then the manifold becomes a perfect fluid spacetime. Finally, we apply this connection to investigate Ricci solitons.

We explore three versions of the Laplacian coflow of $G_2$-structures on circle fibrations over Calabi–Yau 3-folds, interpreting their dimensional reductions to the Kähler geometry of the base. Precisely, we reduce Ansätze for the Laplacian coflow, modified or not by de Turck's trick, both on trivial products $C{Y}^3\times S^1$ and on contact Calabi–Yau 7-manifolds, obtaining in each case a natural modification of the Kähler–Ricci flow.

We study some properties of a $3$-dimensional manifold with a diagonal Riemannian metric as an almost $\eta$-Ricci soliton from the following points of view: under certain assumptions, we determine the potential vector field if $\eta$ is given; we get constraints on the metric when the potential vector field has a particular expression; we compute the defining functions of the soliton when both the potential vector field and the $1$-form are prescribed. Moreover, we find conditions for the manifold to be flat. Based on the theoretical results, we provide examples.

In this paper, we prove that on a smooth Kähler manifold, the $\mathbb{G}$-coercivity of the weighted Mabuchi functional implies the existence of the (v, w)-weighted-cscK (extremal) metric with v log-concave (firstly studied in), e.g, cscK metrics, Kähler-Ricci solitons, $\mu$-cscK metrics.

In this paper, we show the regularity and uniqueness of the twisted conical Kähler-Ricci flow running from a positive closed current with zero Lelong number, which extends the regularizing property of the smooth twisted Kähler-Ricci flow, known as Guedj-Zeriahi's existence theorem and Di Nezza-Lu's uniqueness theorem, to the conical singularity case.

We give a result estimating the dimension of the Lie algebra of Killing vector fields on an irreducible non-trivial gradient Ricci soliton. Then we study the structure of this manifold when the maximal dimension is attained. There are local and global implications.

In this paper, we study the limit behavior of the conical Kähler-Ricci flow as its cone angle tends to zero. More precisely, we prove that as the cone angle tends to zero, the conical Kähler-Ricci flow converges to a unique Kähler-Ricci flow, which is smooth outside the divisor and admits cusp singularity along the divisor.

In this paper we prove localised weighted curvature integral estimates for solutions to the Ricci flow in the setting of a smooth four dimensional Ricci flow or a closed $n$-dimensional Kähler Ricci flow. These integral estimates improve and extend the integral curvature estimates shown by the second author in an earlier paper. If the scalar curvature is uniformly bounded in the spatial $L^p$ sense for some $p>2,$ then the estimates imply a uniform bound on the spatial $L^2$ norm of the Riemannian curvature tensor. Stronger integral estimates are shown to hold if one further assumes a weak non-inflating condition, or we restrict to closed manifolds.

In this paper, we prove that for any asymptotically conical self-shrinker, there exists an embedded closed hypersurface such that the mean curvature flow starting from it develops a singularity modeled on the given shrinker. The main technique is the Ważewski box argument, used by Stolarski in the proof of the corresponding theorem in the Ricci flow case. As a corollary, our construction, combined with the works of Angenent–Ilmanen–Velázquez and Chodosh–Daniels-Holgate–Schulze, implies the existence of fattening level set flows starting from smooth embedded closed hypersurfaces. These provide examples related to a question asked by Evans–Spruck.

Nilpotent Lie groups with left-invariant metrics provide non-trivial examples of Ricci solitons. One typical example is given by the class of two-step nilpotent Lie algebras obtained from simple directed graphs. In this paper, however, we focus on the use of quivers to construct nilpotent Lie algebras. A quiver is a directed graph that allows loops and multiple arrows between two vertices. Utilizing the concept of paths within quivers, we introduce a method for constructing nilpotent Lie algebras from finite quivers without cycles. We prove that for all these Lie algebras, the corresponding simply-connected nilpotent Lie groups admit left-invariant Ricci solitons. The method we introduce constructs a broad family of Ricci soliton nilmanifolds with arbitrarily high degrees of nilpotency.

Gushel-Mukai manifolds are specific families of $n$-dimensional Fano manifolds of Picard rank $1$ and index $n-2$ where $3\le n\le 6$. A Gushel-Mukai $n$-fold is either ordinary, i.e. a hyperquadric section of a quintic Del Pezzo $(n+1)$-fold, or special, i.e. it admits a double cover over the quintic Del Pezzo $n$-fold branched along an ordinary Gushel-Mukai $(n-1)$-fold. In this paper, we prove that a general special Gushel-Mukai $n$-fold is K-stable for every $3\le n\le 6$. Furthermore, we give a description of the first and last walls of the K-moduli of the pair $(M,cQ)$, where $M$ is the quintic Del Pezzo fourfold (or fivefold) and $Q$ is an ordinary Gushel-Mukai threefold (or fourfold). Besides, we compute $\delta$-invariants of quintic Del Pezzo fourfolds and fivefolds which were shown to be K-unstable by K. Fujita, and show that they admit Kähler-Ricci solitons.

In this paper, we study the rigidity of eigenvalues of shring Ricci solitons. It is known that the drifted Laplacian on shrinking Ricci solitons has discrete spectrum, its eigenvalues have a lower bound and a rigidity result holds. Firstly, we show that if the $n^{\text{th}}$ eigenvalue is close to this lower bound, then the $n$-soliton must be the trivial Gaussian soliton ${\mathbb{R}}^n$. Secondly, we show similar results for the $(n-1{)}^{\text{th}}$ and $(n-2{)}^{\text{th}}$ eigenvalue under a non-collapsing condition. Lastly, we give an alomost rigidity for the $k^{\text{th}}$ eigenvalue with general $k$. Part of our results could be viewed as an soliton (could be noncompact) analog of Theorem 1.1 (which only holds for compact manifolds) in Peterson (Invent. Math. 138 (1999): 1-21).

We survey some recent developments on solutions of the Kähler-Ricci flow on compact Kähler manifolds which exist for all positive times.

We consider the Kähler-Ricci flow on compact Kähler manifolds with semiample canonical bundle and intermediate Kodaira dimension, and show that the flow collapses to a canonical metric on the base of the Iitaka fibration in the locally smooth topology and with bounded Ricci curvature away from the singular fibers. This follows from an asymptotic expansion for the evolving metrics, in the spirit of recent work of the first and third-named authors on collapsing Calabi-Yau metrics, and proves two conjectures of Song and Tian.

By adapting previously known arguments concerning Ricci flow and the c-theorem, we give a direct proof that in a two-dimensional sigma-model with compact target space, scale invariance implies conformal invariance in perturbation theory. This argument, which applies to a general sigma-model constructed with a target space metric and B-field, is in accord with a more general proof in the literature that applies to arbitrary two-dimensional quantum field theories. Models with extended supersymmetry and a B-field are known to provide interesting test cases for the relation between scale invariance and conformal invariance in sigma-model perturbation theory. We give examples showing that in such models, the obstructions to conformal invariance suggested by general arguments can actually occur in models with target spaces that are not compact or complete. Thus compactness of the target space, or at least a suitable condition of completeness, is necessary as well as sufficient to ensure that scale invariance implies conformal invariance in models of this type.

Let $(M,g,\omega ,f,\lambda )$ be a Kähler gradient Ricci soliton in real dimension four. One first observes that it is an integrable Hamiltonian system in a classical sense. Indeed, all known complete examples are toric and the symmetry is intrinsically related to the potential function $f$ and the scalar curvature $\text{SS}$. While another article addresses the case that these functions are functionally dependent, this one considers the independent case. The main result states that the soliton admits a toric action under a generic assumption. That is, one assumes that the system is non-degenerate and the potential function $f$ is proper. Then there is an effective, completely integrable Hamiltonian toric ${\mathbb{T}}^2$- action on $(M,\omega )$.

By Perelman's $\mathcal{L}$-geodesic theory, we study the blow-down solutions on a noncompact $\kappa$-noncollapsed steady gradient Ricci soliton $(M^n,g)$ $(n\ge 4)$ with nonnegative curvature operator and positive Ricci curvature away from a compact set of $M$. We prove that any compact split ancient solution of codimension one from the blow-down of $(M,g)$ is of type I. The result is a generalization of our previous work from $n=4$ to any dimension.

In this paper, we prove some Willmore-type inequalities for closed hypersurfaces in weighted manifolds with nonnegative Bakry-Émery Ricci curvature. In particular, we give a sharp Willmore type inequality in steady gradient Ricci solitons. We also prove a sharp Willmore-like inequality in shrinking gradient Ricci solitons. Moreover, we characterize the equality cases of Willmore-type inequalities. These results can be regarded as weighted versions of Agostiniani-Fogagnolo-Mazzieri's Willmore-type inequality. As applications, we derive some sharp isoperimetric type inequalities in weighted manifolds under the existence assumption of a critical set of weighted isoperimetric functional.

Metric measure spaces with synthetic Ricci bounds have attracted great interest in recent years, accompanied by spectacular breakthroughs and deep new insights. In this survey, I will provide a brief introduction to the concept of lower Ricci bounds as introduced by Lott-Villani and myself, and illustrate some of its geometric, analytic and probabilistic consequences, among them Li-Yau estimates, coupling properties for Brownian motions, sharp functional and isoperimetric inequalities, rigidity results, and structural properties like rectifiability and rectifiability of the boundary. In particular, I will explain its crucial interplay with the heat flow and its link to the curvaturedimension condition formulated in functional-analytic terms by Bakry-Èmery. This equivalence between the Lagrangian and the Eulerian approach then will be further explored in various recent research directions: i) time-dependent Ricci bounds which provide a link to (super-) Ricci flows for singular spaces, ii) second order calculus, upper Ricci bounds, and transformation formulas, iii) distribution-valued Ricci bounds which e.g. allow singular effects of non-convex boundaries to be taken into account.

We develop a framework inspired by Lauret's "bracket flow" to study the generalized Ricci flow, as introduced by Streets, on discrete quotients of Lie groups. As a first application, we establish global existence on solvmanifolds in arbitrary dimensions, a result which is new even for the pluriclosed flow. We also define a notion of generalized Ricci soliton on exact Courant algebroids that is geometrically meaningful and allows for non-trivial expanding examples. On nilmanifolds, we show that these solitons arise as rescaled limits of the generalized Ricci flow, provided the initial metrics have "harmonic torsion", and we classify them in low dimensions. Finally, we provide a new formula for the generalized Ricci curvature of invariant generalized metrics in terms of a moment map for the action of a non-reductive real Lie group.

In this article we derive gradient estimation for positive solution of the equation \beginequation* (\partial_t-Δ_f)u = A(u)p(x,t) + B(u)q(x,t) + \mathcalG(u) \endequation* on a weighted Riemannian manifold evolving along the $(k,m)$ super Perelman-Ricci flow \beginequation* \frac{\partial g}{\partial t}(x,t)+2Ric_f^m(g)(x,t)\ge -2kg(x,t). \endequation* As an application of gradient estimation we derive a Harnack type inequality along with a Liouville type theorem.

Building on works of Boulanger and Goto, we show that Goto's scalar curvature is the moment map for an action of generalized Hamiltonian automorphisms of the associated Courant algebroid, constrained by the choice of an adapted volume form. We derive an explicit formula for Goto's scalar curvature, and show that it is constant for generalized Kähler-Ricci solitons. Restricting to the generically symplectic type case, we realize the generalized Kähler class as the complexified orbit of the Hamiltonian action above. This leads to a natural extension of Mabuchi's metric and $K$-energy, implying a conditional uniqueness result. Finally, in this setting we derive a Calabi-Matsushima-Lichnerowicz obstruction and a Futaki invariant.

Deep neural networks (DNNs) are powerful tools for approximating the distribution of complex data. It is known that data passing through a trained DNN classifier undergoes a series of geometric and topological simplifications. While some progress has been made toward understanding these transformations in neural networks with smooth activation functions, an understanding in the more general setting of non-smooth activation functions, such as the rectified linear unit (ReLU), which tend to perform better, is required. Here we propose that the geometric transformations performed by DNNs during classification tasks have parallels to those expected under Hamilton's Ricci flow - a tool from differential geometry that evolves a manifold by smoothing its curvature, in order to identify its topology. To illustrate this idea, we present a computational framework to quantify the geometric changes that occur as data passes through successive layers of a DNN, and use this framework to motivate a notion of `global Ricci network flow' that can be used to assess a DNN's ability to disentangle complex data geometries to solve classification problems. By training more than $1,500$ DNN classifiers of different widths and depths on synthetic and real-world data, we show that the strength of global Ricci network flow-like behaviour correlates with accuracy for well-trained DNNs, independently of depth, width and data set. Our findings motivate the use of tools from differential and discrete geometry to the problem of explainability in deep learning.

Motivated by recent work of Deruelle-Schulze-Simon, we study complete weakly PIC1 Ricci flows with Euclidean volume growth coming out of metric cones. We show that such a Ricci flow must be an expanding gradient Ricci soliton, and as a consequence, any metric cone at infinity of a complete weakly PIC1 Kähler manifold with Euclidean volume growth is biholomorphic to complex Euclidean space in a canonical way.

Based on the renormalization group (RG) flow of worldsheet bosonic string theory, we construct an effective holographic dual description of the target space theory identifying the RG scale with the emergent extra dimension. This results in an effective dilaton-gravity-gauge theory, analogous to the low-energy description of bosonic M-theory. We argue that this holographic dual effective field theory is non-perturbative in the ${\alpha }^{\prime }$ expansion, where a class of string quantum fluctuations are resummed to all orders. To investigate the monotonicity of the RG flow of the target space metric in the emergent spacetime, we consider entropy production along the RG flow. We construct a microscopic entropy functional based on the probability distribution function of the holographic dual effective field theory, regarded as Gibbs- or Shannon-type entropy. Given that the Ricci flow represents the 1-loop RG flow equation of the target space metric for the 2D non-linear sigma model, and motivated by Perelman's proof of the monotonicity of Ricci flow, we propose a Perelman's entropy functional for the holographic dual effective field theory. This entropy functional is also non-perturbative in the ${\alpha }^{\prime }$ expansion, and thus, generalizes the 1-loop result to the all-loop order. Furthermore, utilizing the equivalence between the Hamilton-Jacobi equation and the local RG equation, we suggest that the RG flow of holographic Perelman's entropy functional is the Weyl anomaly. This eventually reaffirms the monotonicity of RG flow for the emergent target spacetime but in a non-perturbative way. Interestingly, we find that the microscopic entropy production rate can be determined by integrating the rate of change of the holographic Perelman's entropy functional over all possible metric configurations along the flow.

We study the behavior of the Cheeger isoperimetric constant under the Ricci flow on compact surfaces. For metrics on a surface diffeomorphic to $S^2$, we show that the Cheeger constant is non-decreasing along the flow. The proof uses evolution identities for parallel curves together with a viscosity formulation of the evolution of $\log h$ which accommodates for the possible switching of minimizing regions. We also give examples of nontrivial Ricci flows on topological $2$-spheres for which the Cheeger constant remains constant, demonstrating that strict monotonicity is not expected.

We establish three circles theorems for subharmonic functions on Riemannian manifolds with nonnegative Ricci curvature, as well as on gradient shrinking Ricci solitons with scalar curvature bounded from below by $\frac{n-2}{2}$. We also establish a three circiles theorem for holomorphic functions on gradient shrinking Kähler-Ricci solitons with some curvature conditions. As applications, we prove some Liouville type theorems.

We show that every gradient shrinking soliton of the generalized Ricci flow on compact manifold is a Ricci soliton. And we prove that the pluriclosed soliton is gradient Kahler-Ricci soliton under a broad cohomological condition. Moreover, we construct the first example of non-trivial shrinking generalized soliton, which can serve as a singularity model of the generalized Ricci flow.

In this paper, we study the existence and rigidity of (degenerated) circle pattern metric with prescribed total geodesic curvatures in spherical background geometry. To find the (degenerated) circle pattern metric with prescribed total geodesic curvatures, we define some prescribed combinatorial Ricci flows and study the convergence of flows for (degenerated) circle pattern metrics. We solve the prescribed total geodesic curvature problem and provide two methods to find the degenerated circle pattern metric with prescribed total geodesic curvatures. As far as we know, this is the first degenerated result for total geodesic curvatures in spherical background geometry.

We classify superpotentials for the Hamiltonian system corresponding to the cohomogeneity one gradient Ricci soliton equations. Aside from recovering known examples of superpotentials for steady solitons, we find a new superpotential on a specific case of the Bérard Bergery-Calabi ansatz. The latter is used to obtain an explicit formula for a steady complete soliton with an equidistant family of hypersurfaces given by circle bundles over $S^2\times S^2$. There are no superpotentials in the non-steady case in dimensions greater than 2, even if polynomial coefficients are allowed. We also briefly discuss generalised first integrals and the limitations of some known methods of finding them.

In this note, we provide a very simple proof of the uniformization theorem of Riemann surfaces by Ricci flow. The argument builds on a refinement of Hamilton's isoperimetric estimate for the Ricci flow on the two-sphere.

We establish a transcendental generalization of Nakamaye's theorem to compact complex manifolds when the form is not assumed to be closed. We apply the recent analytic technique developed by Collins and Tosatti to show that the non-Hermitian locus of a nef and big $(1,1)$-form, which is not necessarily closed, on a compact complex manifold equals the union of all positive-dimensional analytic subvarieties where the restriction of the form is not big (null locus). As an application, we can give an alternative proof of the Nakai–Moishezon criterion of Buchdahl and Lamari for complex surfaces and generalize this result in higher dimensions Finally, we investigate finite time non-collapsing singularities of the Chern–Ricci flow, partially answering a question raised by Tosatti and Weinkove.

We study the existence and small scale behaviour of almost splitting maps along a Ricci flow satisfying Type I curvature bounds. These are special solutions of the heat equation that serve as parabolic analogues of harmonic almost splitting maps, which have proven to be an indespensable tool in the study of the structure of the singular set of non-collapsed Ricci limit spaces. In this paper, motivated by the recent work of Cheeger-Jiang-Naber in the Ricci limit setting, we construct sharp splitting maps on Ricci flows that are almost selfsimilar, and then investigate their small scale behaviour. We show that, modulo linear transformations, an almost splitting map at a large scale remains a splitting map even at smaller scales, provided that the Ricci flow remains sufficiently self-similar. Allowing these linear transformations means that a priori an almost splitting map might degenerate at small scales. However, we show that under an additional summability hypothesis such degeneration doesn't occur.

This is a sequel to our paper [24], in which we investigated the geometry of 4-dimensional gradient shrinking Ricci solitons with half positive (nonnegative) isotropic curvature. In this paper, we mainly focus on 4-dimensional gradient steady Ricci solitons with nonnegative isotropic curvature (WPIC) or half nonnegative isotropic curvature (half WPIC). In particular, for 4D complete ancient solutions with WPIC, we are able to prove the 2-nonnegativity of the Ricci curvature and bound the curvature tensor Rm by |Rm|\leq R. For 4D gradient steady solitons with WPIC, we obtain a classification result. We also give a partial classification of 4D gradient steady Ricci solitons with half WPIC. Moreover, we obtain a preliminary classification result for 4D complete gradient expanding Ricci solitons with WPIC. Finally, motivated by the recent work [59], we improve our earlier results in [24] on 4D gradient shrinking Ricci solitons with half PIC or half WPIC, and also provide a characterization of complete gradient Kaehler-Ricci shrinkers in complex dimension two among 4-dimensional gradient Ricci shrinkers.

We investigate transverse Ricci solitons, the self-similar solutions of the transverse Ricci flow, on a compact foliated manifold. In particular, we show the relations between a taut Riemannian foliation and a transverse Ricci soliton. Moreover, we find some examples of transverse Ricci solitons.

We prove that there exist ${\mathsf{SU}}_3$-invariant metrics on Aloff-Wallach spaces $W_{k_1,k_2}^7$, as well as ${\mathsf{SU}}_5$-invariant metrics on the Berger space $B^{13}$, which have positive sectional curvature and evolve under the Ricci flow to metrics with non-positively curved planes.

Let $P$ be a point of a compact Riemann surface $X$. We study self-adjoint extensions of the Dolbeault Laplacians in hermitian line bundles $L$ over $X$ initially defined on sections with compact supports in $X\backslash \{P\}$. We define the $\zeta$-regularized determinants for these operators and derive comparison formulas for them. We introduce the notion of the Robin mass of $L$. This quantity enters the comparison formulas for determinants and is related to the regularized $\zeta (1)$ for the Dolbeault Laplacian. For spinor bundles of even characteristic, we find an explicit expression for the Robin mass. In addition, we propose an explicit formula for the Robin mass in the scalar case. Using this formula, we describe the evolution of the regularized $\zeta (1)$ for scalar Laplacian under the Ricci flow. As a byproduct, we find an alternative proof for the Morpurgo result that the round metric minimizes the regularized $\zeta (1)$ for surfaces of genus zero.

If we want to deform a compact Riemannian manifold with boundary using Ricci flow, we first need to decide on appropriate boundary conditions. We would like these conditions to reflect the geometric nature of the flow and allow for a variety of initial data. Importantly, the conditions should be compatible with the expected evolution of Einstein metrics. We propose it is natural to choose those conditions, for which the first variation of certain functionals, such as the Einstein-Hilbert action and Perelmans lambda-functional, does not admit a boundary term. We provide a proof of the short term existence of solutions of the initial boundary value problem, under these conditions.

Kröncke has shown that the Fubini-Study metric is an unstable generalized stationary solution of Ricci flow [Krö20]. In this paper, we carry out numerical simulations which indicate that Ricci flow solutions originating at unstable perturbations of the Fubini-Study metric develop local singularities modeled by the blowdown soliton discovered in [FIK03].

In $1996$, H.-D. Cao constructed a $U(n)$-invariant steady gradient Kähler-Ricci soliton on ${\mathbb{C}}^n$ and asked whether every steady gradient Kähler-Ricci soliton of positive curvature on ${\mathbb{C}}^n$ is necessarily $U(n)$-invariant (and hence unique up to scaling). Recently, Apostolov-Cifarelli answered this question in the negative for $n=2$. Here, we construct a family of $U(1)\times U(n-1)$-invariant, but not $U(n)$-invariant, complete steady gradient Kähler-Ricci solitons with strictly positive curvature operator on real $(1,1)$-forms (in particular, with strictly positive sectional curvature) on ${\mathbb{C}}^n$ for $n\ge 3$, thereby answering Cao's question in the negative for $n\ge 3$. This family of steady Ricci solitons interpolates between Cao's $U(n)$-invariant steady Kähler-Ricci soliton and the product of the cigar soliton and Cao's $U(n-1)$-invariant steady Kähler-Ricci soliton. This provides the Kähler analog of the Riemannian flying wings construction of Lai. In the process of the proof, we also demonstrate that the almost diameter rigidity of ${\mathbb{P}}^n$ endowed with the Fubini-Study metric does not hold even if the curvature operator is bounded below by $2$ on real $(1,1)$-forms.

We prove long-time existence of the Ricci flow starting from complete manifolds with bounded curvature and scale-invariant integral curvature sufficiently pinched with respect to the inverse of its Sobolev constant. Moreover, if the curvature is sub-critical $L^p$-integrable, this flow converges locally smoothly to a limiting metric $g(\infty )$ on $M$ with $(M,g(\infty ))$ isometric to the standard flat ${\mathbb{R}}^n$, which implies topological rigidity of $M$. This generalizes work of Chen, who proved analogous results for asymptotically flat manifolds. We also prove a long-time Ricci flow existence (and likewise topological rigidity) result for unbounded curvature initial data, assuming the initial data is a locally smooth limit of bounded curvature manifolds as described above.

In this paper, (gradient) almost Ricci solitons on Finsler measure spaces $(M,F,m)$ are introduced and investigated. We prove that $(M,F,m)$ is a gradient almost Ricci soliton if and only if the infinity-Ricci curvature Ric${}_{\infty }$ is a scalar function on $M$ when $M$ is compact. Moreover, we give an equivalent characterization of (gradient) almost Ricci solitons for Randers metrics $F=\alpha +\beta$, which implies that every Randers (gradient) almost Ricci soliton is of isotropic S${}_{BH}$-curvature. Based on this and the navigation technique, we further classify Randers almost Ricci solitons (resp. gradient almost Ricci solitons) up to classifications of Randers Einstein metrics $F$ (resp. Riemannian gradient almost Ricci solitons) and the homothetic vector fields of $F$ (resp. solutions of the equation which the weight function $f$ of $m$ satisfies) when $F$ has isotropic S${}_{BH}$-curvature. As applications, we obtain some rigidity results for compact Randers (gradient) Ricci solitons and construct several Randers gradient Ricci solitons, which are the first nontrivial examples of gradient Ricci solitons in Finsler geometry.

In this paper, we introduce a new notion of curvature on the edges of a graph that is defined in terms of effective resistances. We call this the Ricci–Foster curvature. We study the Ricci flow resulting from this curvature. We prove the existence of solutions to Ricci flow on short time intervals, and prove that Ricci flow preserves graphs with nonnegative (resp. positive) curvature.

We study $n$-dimensional Ricci flows with non-negative Ricci curvature where the curvature is pointwise controlled by the scalar curvature and bounded by $C/t$, starting at metric cones which are Reifenberg outside the tip. We show that any such flow behaves like a self-similar solution up to an exponential error in time. As an application, we show that smooth $n$-dimensional complete non-compact Riemannian manifolds which are uniformly PIC1-pinched, with positive asymptotic volume ratio, are Euclidean. This confirms a higher dimensional version of a conjecture of Hamilton and Lott under the assumption of non-collapsing. It also yields a new and more direct proof of the original conjecture of Hamilton and Lott in three dimensions.

We extend the anholonomic frame and connection deformation method, AFCDM, for constructing exact and parametric solutions in general relativity, GR, to geometric flow models and modified gravity theories, MGTs, with nontrivial torsion and nonmetricity fields. Following abstract geometric or variational methods, we can derive corresponding systems of nonmetric gravitational and matter field equations which consist of very sophisticated systems of coupled nonlinear PDEs. Using nonholonomic frames with dyadic spacetime splitting and applying the AFCDM, we prove that such systems of PDEs can be decoupled and integrated in general forms for generic off-diagonal metric structures and generalized affine connections. We generate new classes of quasi-stationary solutions (which do not depend on time like coordinates) and study the physical properties of some physically important examples. Such exact or parametric solutions are determined by nonmetric solitonic distributions and/or ellipsoidal deformations of wormhole hole configurations. It is not possible to describe the thermodynamic properties of such solutions in the framework of the Bekenstein-Hawking paradigm because such metrics do not involve, in general, certain horizons, duality, or holographic configurations. Nevertheless, we can always elaborate on associated Grigori Perelman thermodynamic models elaborated for nonmetric geometric flows. In explicit form, applying the AFCDM, we construct and study the physical implications of new classes of traversable wormhole solutions describing solitonic deformation and dissipation of non-Riemannian geometric objects. Such models with nontrivial gravitational off-diagonal vacuum are important for elaborating models of dark energy and dark matter involving wormhole configurations and solitonic-type structure formation.

This is a survey on the Strominger system and a geometric flow known as the anomaly flow. We will discuss various aspects of non-Kähler geometry on Calabi-Yau threefolds. Along the way, we discuss balanced metrics and balanced classes, the Aeppli cohomology class associated to a solution to the Strominger system, the equations of motion of heterotic supergravity, and a version of Ricci flow in this special geometry.

This article carries out the investigation of a three-dimensional Riemannian manifold $N^3$ endowed with a semi-symmetric type non-metric connection. Firstly, we construct a non-trivial example to prove the existence of a semi-symmetric type non-metric connection on $N^3$. It is established that a $N^3$ with the semi-symmetric type non-metric connection, whose metric is a gradient Ricci soliton, is a manifold of constant sectional curvature with respect to the semi-symmetric type non-metric connection. Moreover, we prove that if the Riemannian metric of $N^3$ with the semi-symmetric type non-metric connection is a gradient Yamabe soliton, then either $N^3$ is a manifold of constant scalar curvature or the gradient Yamabe soliton is trivial with respect to the semi-symmetric type non-metric connection. We also characterize the manifold $N^3$ with a semi-symmetric type non-metric connection whose metrics are Einstein solitons and $m$-quasi Einstein solitons of gradient type, respectively.

In this article, we examine gradient type Ricci solitons and $(m,\tau )$-quasi Einstein solitons in generalized Robertson-Walker ($GRW$) spacetimes. Besides, we demonstrate that in this scenario the $GRW$ spacetime presents the Robertson-Walker ($RW$) spacetime and the perfect fluid ($PF$) spacetime presents the phantom era. Consequently, we show that if a $GRW$ spacetime permits a gradient $\tau$- Einstein solitons, then it also represents a $PF$ spacetime under certain condition.

Machine learning techniques are increasingly powerful, leading to many breakthroughs in the natural sciences, but they are often stochastic, error-prone, and blackbox. How, then, should they be utilized in fields such as theoretical physics and pure mathematics that place a premium on rigor and understanding? In this Perspective we discuss techniques for obtaining rigor in the natural sciences with machine learning. Non-rigorous methods may lead to rigorous results via conjecture generation or verification by reinforcement learning. We survey applications of these techniques-for-rigor ranging from string theory to the smooth $4$d Poincaré conjecture in low-dimensional topology. One can also imagine building direct bridges between machine learning theory and either mathematics or theoretical physics. As examples, we describe a new approach to field theory motivated by neural network theory, and a theory of Riemannian metric flows induced by neural network gradient descent, which encompasses Perelman's formulation of the Ricci flow that was utilized to resolve the $3$d Poincaré conjecture.

Let $(M,g)$ be a closed connected oriented (possibly non-spin) smooth four-dimensional manifold with scalar curvature bounded below by $n(n-1)$. In this paper, we prove that if $f$ is a smooth map of non-zero degree from $(M,g)$ to the unit four-sphere, then $f$ is an isometry. Following ideas of Gromov, we use $\mu$-bubbles and a version with coefficients of the rigidity of the three-sphere to rule out the case of strict inequality. Our proof of rigidity is based on the harmonic map heat flow coupled with the Ricci flow.

Sets related to positively curved invariant Riemannian metrics on generalized Wallach spaces are considered. The problem arises in studying of the evolution of such metrics under the normalized Ricci flow equation. For Riemannian metrics of the Wallach spaces $\mathrm{SU}(3)/T_{\max }$, $Sp(3)/{\left(Sp(1)\right)}^3$ and $F_4/Spin(8)$ which admit positive sectional curvature and belong to a given invariant surface $\Sigma$ of the normalized Ricci flow we established that they form a set bounded by three connected and pairwise disjoint regular space curves such that each of them approaches two others asymptotically at infinity. Analogously, for all generalized Wallach spaces the set of Riemannian metrics which belong to $\Sigma$ and admit positive Ricci curvature is bounded by three curves each consisting of two connected components as regular curves. Intersections and asymptotical behaviors of these components were studied as well.

We develop a theory of Ricci flow for metrics on Courant algebroids which unifies and extends the analytic theory of various geometric flows, yielding a general tool for constructing solutions to supergravity equations. We prove short time existence and uniqueness of solutions on compact manifolds, in turn showing that the Courant isometry group is preserved by the flow. We show a scalar curvature monotonicity formula and prove that generalized Ricci flow is a gradient flow, extending fundamental works of Hamilton and Perelman. Using these we show a convergence result for certain nonsingular solutions to generalized Ricci flow.

This article consists of an introduction to the theory of nonassociative geometric classical and quantum information flows defined by star products with R-flux deformations in string gravity. Corresponding nonassociative generalizations of the concepts of classical Shannon entropy, quantum von Neumann entropy, Rényi entropy are formulated. The fundamental geometric and quantum information objects are computed following the Grigori Perelman statistical thermodynamic approach to Ricci flows and gravity theories generalized for phase spaces modelled as (co) tangent Lorentz bundles. Nonassociative parametric deformations and nonholonomic thermo-geometric versions of statistical generating functions, their quantum analogues as density matrices are considered for deriving the entropy, energy and fluctuation functionals. This allows us to define and compute respective classical and quantum relative and conditional entropies, mutual information and nonassociative entanglement and thermodynamic information variables. We formulate the principles of nonassociative quantum geometric and information flow theory, QGIF, and study the basic properties of such quasi-stationary models related to modified gravity theories. Applications are considered for nonassociative deformed and entangled couples of four-dimensional, 4-d, wormholes (defined by respective spacetime and/or momentum type coordinates) and nonassociative QGIFs of 8-d phase space generalized wormholes configurations. Finally, we speculate on phase space black holes and wormholes being transversable for nonassociative qubits, quantum channels and entanglement witness; thought and laboratory experiments are discussed; and perspectives for quantum computer modelling and tests of nonassociative geometric flow and gravity theories are considered.

In this paper, we study the Bakry-Émery Ricci flow on finite graphs. Our main result is the local existence and uniqueness of solutions to the Ricci flow. We prove the long-time convergence or finite-time blow up for the Bakry-Émery Ricci flow on finite trees and circles.

We generalize Kähler-Ricci solitons to the almost-Kähler setting as the zeros of Inoue's moment map, and show that their existence is an obstruction to the existence of first-Chern-Einstein almost-Kähler metrics on compact symplectic Fano manifolds. We prove deformation results of such metrics in the $4$-dimensional case. Moreover, we study the Lie algebra of holomorphic vector fields on $2n$-dimensional compact symplectic Fano manifolds admitting generalized almost-Kähler-Ricci solitons. In particular, we partially extend Matsushima's theorem to compact first-Chern-Einstein almost-Kähler manifolds.

We prove an $L^2$ estimate for the drift heat equation on a complete gradient shrinking Ricci soliton. This estimate has a time-dependent weight which is Gaussian in its spatial asymptotics. When transferred and scaled to an estimate for the heat equation along the Ricci flow of the soliton, this estimate is uniform up to the singular time.

Let $(M^n,g)$ $(n\ge 4)$ be a complete noncompact $\kappa$-noncollapsed steady Ricci soliton with $\mathrm{Rm}\ge 0$ and $\mathrm{Ric}>0$ away from a compact set $K$ of $M$. We prove that there is no any $(n-1)$-dimensional compact split limit Ricci flow of type I arising from the blow-down of $(M,g)$, if there is an $(n-1)$-dimensional noncompact split limit Ricci flow. Consequently, the compact split limit ancient flows of type I and type II cannot occur simultaneously from the blow-down. As an application, we prove that $(M^n,g)$ with $\mathrm{Rm}\ge 0$ must be isometric the Bryant Ricci soliton up to scaling, if there exists a sequence of rescaled Ricci flows $(M,g_{p_i}(t);p_i)$ of $(M,g)$ converges subsequently to a family of shrinking quotient cylinders.

We explicitly determine the optimal degenerations of Fano threefolds $X$ in family No 2.23 of Mori-Mukai's list as predicted by the Hamilton-Tian conjecture. More precisely, we find a special degeneration $(\mathcal{X},{\xi }_0)$ of $X$ such that $({\mathcal{X}}_0,{\xi }_0)$ is weighted K-polystable, which is equivalent to $({\mathcal{X}}_0,{\xi }_0)$ admitting a Kähler-Ricci soliton (KRS) by and. Furthermore, we study the moduli spaces of $({\mathcal{X}}_0,{\xi }_0)$. The $\mathbf{H}$-invariant of $X$ divides the natural parameter space into two strata, which leads to different moduli spaces of KRS Fano varieties. We show that one of them is isomorphic to the GIT-moduli space of biconic curves $C\subseteq {\mathbb{P}}^1\times {\mathbb{P}}^1$, and the other one is a single point.

Wrinkling instabilities of thin elastic sheets can be used to generate periodic structures over a wide range of length scales. Viscosity of the thin elastic sheet or its surrounding medium has been shown to be responsible for dynamic processes. While this has been explored for solid as well as liquid thin elastic sheets we here consider wrinkling of fluid deformable surfaces, which show a solid-fluid duality and have been established as model systems for biomembranes and cellular sheets. We use this hydrodynamic theory and numerically explore the formation of wrinkles and their coarsening, either by a continuous reduction of the enclosed volume or the continuous increase of the surface area. Both lead to almost identical results for wrinkle formation and the coarsening process, for which a universal scaling law for the wavenumber is obtained for a broad range of surface viscosity and rate of change of volume or area. However, for large Reynolds numbers and small changes in volume or area wrinkling can be suppressed and surface hydrodynamics allows for global shape changes following the minimal energy configurations of the Helfrich energy for corresponding reduced volumes.

The primary focus of the current study is to explore the geometrical properties of the Vaidya-Bonner-de Sitter (briefly, VBdS) spacetime, which is a generalization of Vaidya-Bonner spacetime, Vaidya spacetime and Schwarzschild spacetime. In this study we have shown that the VBdS spacetime describes various types of pseudosymmetric structures, including pseudosymmetry due to conformal curvature, conharmonic curvature and other curvatures. Additionally, it is shown that such a spacetime is 2-quasi-Einstein, Einstein manifold of level 3, generalized Roter type, and that conformal 2-forms are recurrent. The geometric features of the Vaidya-Bonner spacetime, Vaidya spacetime, and Schwarzschild spacetime are obtained as a particular instance of the main determination. It is further established that the VBdS spacetime admits almost Ricci soliton and almost η-Yamabe soliton with respect to non-Killing vector fields. Also, it is proved that such a spacetime possesses generalized conharmonic curvature inheritance. It is interesting to note that in the VBdS spacetime the tensors Q(T,R), Q(S,R) and Q(g,R) are linearly dependent. Finally, this spacetime is compared with the Vaidya-Bonner spacetime with respect to their admitting geometric structures, viz., various kinds of symmetry and pseudosymmetry properties.

In this paper, we study a natural discretization of the smooth Gaussian curvature on surfaces. A discrete uniformization theorem is established for this discrete Gaussian curvature. We further investigate the prescribing combinatorial curvature problem for a parametrization of this discrete Gaussian curvature, which is called the combinatorial $\alpha$-curvature. To find decorated piecewise hyperbolic metrics with prescribed combinatorial $\alpha$-curvatures, we introduce the combinatorial $\alpha$-Ricci flow for decorated piecewise hyperbolic metrics. To handle the potential singularities along the combinatorial $\alpha$-Ricci flow, we do surgery along the flow by edge flipping under the weighted Delaunay condition. Then we prove the longtime existence and convergence of the combinatorial $\alpha$-Ricci flow with surgery. As an application of the combinatorial $\alpha$-Ricci flow with surgery, we give the existence of decorated piecewise hyperbolic metrics with prescribed combinatorial $\alpha$-curvatures. We further introduce the combinatorial $\alpha$-Calabi flow with surgery and study its longtime behavior.

We prove the existence of a one-parameter family of pairwise non-isometric, complete, positively curved, steady generalized Ricci solitons of gradient type on ${\mathbb{R}}^3$ that are invariant under the natural cohomogeneity one action of SO(3). In the context of generalized Ricci flow, this result represents the analogue of Bryant's construction of the complete rotationally invariant steady soliton for the Ricci flow.

We prove the continuity of bounded solutions to complex Monge-Ampère equations on reduced, locally irreducible compact Kähler spaces. This in particular implies that any singular Kähler-Einstein potentials constructed in and are continuous. We also provide an affirmative answer to a conjecture in by showing that a resolution of any compact normal Kähler space satisfies the continuous approximation property. Finally, we settle the continuity of the potentials of the weak Kähler-Ricci flows on compact Kähler varieties with log terminal singularities.

A noncollapsed $\mathbb{F}$-limit metric soliton is a self-similar singularity model that inevitably arises when studying the Ricci flow with the tool of $\mathbb{F}$-convergence [Bam20a,Bam20b,Bam20c]. In this article, we shall present a systematic study of the noncollapsed $\mathbb{F}$-limit metric soliton, and show that, apart from the known results in [Bam20c], it satisfies many properties of smooth Ricci shrinkers. In particular, we show a quadratic lower bound for the scalar curvature, a local gap theorem, a global Sobolev inequality, and an optimal volume growth lower bound.

The fixed points of the generalized Ricci flow are the Bismut Ricci flat metrics, i.e., a generalized metric $(g,H)$ on a manifold $M$, where $g$ is a Riemannian metric and $H$ a closed $3$-form, such that $H$ is $g$-harmonic and $\mathrm{Rc}(g)=\frac{1}{4}{H}_g^2$. Given two standard Einstein homogeneous spaces $G_i/K$, where each $G_i$ is a compact simple Lie group and $K$ is a closed subgroup of them holding some extra assumption, we consider $M=G_1\times G_2/\Delta K$. Recently, Lauret and Will proved the existence of a Bismut Ricci flat metric on any of these spaces. We proved that this metric is always asymptotically stable for the generalized Ricci flow on $M$ among a subset of $G$-invariant metrics and, if $G_1=G_2$, then it is globally stable.

In this paper, we investigate homogeneous Riemannian geometry on real flag manifolds of the split real form of ${\mathfrak{g}}_2$. We characterize the metrics that are invariant under the action of a maximal compact subgroup of $G_2.$ Our exploration encompasses the analysis of g.o. metrics and equigeodesics on the ${\mathfrak{g}}_2$-type flag manifolds. Additionally, we explore the Ricci flow for the case where the isotropy representation has no equivalent summands, employing techniques from the qualitative theory of dynamical systems.

We study polynomial growth holomorphic functions and forms on complete gradient shrinking Ricci solitons. By relating to the spectral data of the $f$-Laplacian, we show that the dimension of the space of polynomial growth holomorphic functions or holomorphic $(p,0)$-forms are finite. In particular, a sharp dimension estimate for the space of linear growth holomorphic functions was obtained. Under some additional curvature assumption, we prove an almost sharp estimate for the frequency of polynomial growth holomorphic functions, which was used to obtain dimension upper bound as a power function of the polynomial order.

In this paper, we construct solutions of Lagrangian mean curvature flow which exist and are embedded for all time, but form an infinite-time singularity and converge to an immersed special Lagrangian as $t\to \infty$. In particular, the flow decomposes the initial data into a union of special Lagrangians intersecting at one point. This result shows that infinite-time singularities can form in the Thomas–Yau `semi-stable' situation. A precise polynomial blow-up rate of the second fundamental form is also shown. The infinite-time singularity formation is obtained by a perturbation of an approximate family $N^{\epsilon (t)}$ constructed by gluing in special Lagrangian `Lawlor necks' of size $\epsilon (t)$, where the dynamics of the neck size $\epsilon (t)$ are driven by the obstruction for the existence of nearby special Lagrangians to $N^{\epsilon (t)}$. This is inspired by the work of Brendle and Kapouleas regarding ancient solutions of the Ricci flow.

For a Fano manifold, We consider the geometric quantization of the Kähler-Ricci flow and the associated entropy functional. Convergence to the original flow and entropy is established. It is also possible to formulate the finite-dimensional analogue of the optimal degeneration for the anti-canonical polarization.

The aim of this article is to explore the Clairaut anti-invariant Riemannian maps from/to Kähler manifolds admitting Ricci solitons. We find the curvature relations and calculate the Ricci tensor under different conditions. We discuss the condition under which range space becomes $\alpha$-Ricci soliton. We obtain conditions for the range and kernel spaces of these maps to be Einstein. Next, we find the scalar curvature for range space. Further, we give the relation between Ricci curvature and Lie derivative under these maps. Moreover, we find the condition for a vertical potential vector field on target manifold to be conformal vector field on range space of these maps. Finally, we give non-trivial examples of such maps.