Papers from 2023

We prove that a complete solution to the Ricci flow on $M\times [-T,0)$ which has quadratic curvature decay on some end of $M$ and converges locally smoothly to the end of a cone on that neighborhood as $t\nearrow 0$ must be a gradient shrinking soliton.

We establish sufficient conditions which ensure that a locally-warped product structure propagates backward in time under the Ricci flow. As an application, we prove that if an asymptotically conical gradient shrinking soliton is asymptotic to a cone whose cross-section is a product of Einstein manifolds, the soliton must itself be a multiply-warped product over the same manifolds.

We show that the set of awesome homogeneous metrics on non-compact manifolds is Ricci flow invariant. Moreover, if the universal cover of such awesome homogeneous space is not contractible the Ricci flow has finite extinction time, confirming the Dynamical Alekseevskii Conjecture in this case. We also analyze the long-time limits of awesome homogeneous Ricci flows.

We prove dynamical stability and instability theorems for Poincaré-Einstein metrics under the Ricci flow. Our key tool is a variant of the expander entropy for asymptotically hyperbolic manifolds, which Dahl, McCormick and the first author established in a recent article. It allows us to characterize stability and instability in terms of a local positive mass theorem and in terms of volume comparison for nearby metrics.

In this work, we consider the area non-increasing map between manifolds with positive curvature. By exploring the strong maximum principle along the graphical mean curvature flow, we show that an area non-increasing map between certain positively curved manifolds is either homotopy trivial, Riemannian submersion, local isometry or isometric immersion. This implies that an area non-increasing self map of ${\mathbb{CP}}^n$, $n\ge 2$ is either homotopically trivial or is an isometry. This confirms a speculation of Tsai-Tsui-Wang. We also use Brendle's sphere Theorem and mean curvature flow coupled with Ricci flow to establish related results on manifolds with positive $1$-isotropic curvature.

We study the behavior of a three-dimensional dynamical system with respect to some set $S$ given in 3-dimensional euclidian space. Geometrically such a system arises from the normalized Ricci flow on some class of generalized Wallach spaces that can be described by a real parameter $a\in (0,1/2)$, as for $S$ it represents the set of invariant Riemannian metrics of positive sectional curvature on the Wallach spaces. Establishing that $S$ is bounded by three conic surfaces and regarding the normalized Ricci flow as an abstract dynamical system we find out the character of interrelations between that system and $S$ for all $a\in (0,1/2)$. These results can cover some well-known results, in particular, they can imply that the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curvature into metrics with mixed sectional curvature on the Wallach spaces corresponding to the cases $a\in \{1/9,1/8,1/6\}$ of generalized Wallach spaces.

In this paper, we explore the positive solutions to the Finslerian nonlinear equation $$\frac{\partial u}{\partial t}={\Delta }^{\nabla u}u+au\log u+bu,$$ which is related to Ricci solitons and serves as the Euler-Lagrange equation to the Finslerian log-energy functional. We then obtain the global gradient estimate of its positive solution on a compact Finsler metric measure space with the weighted Ricci curvature bounded below. Furthermore, using a new comparison theorem developed by the first author, we also establish a local gradient estimate on a non-compact forward complete Finsler metric measure spaces with the mixed weighted Ricci curvature bounded below, as well as finite bounds of misalignment and some non-Riemannian curvatures. Lastly, we prove the Harnack inequalities and a Liouville-type theorem of such solutions.

Giving explicit parametrizations of discrete constant Gaussian curvature surfaces of revolution that are defined from an integrable systems approach, we study Ricci flow for discrete surfaces, and see how discrete surfaces of revolution have a geometric realization for the Ricci flow that approaches the constant Gaussian curvature surfaces we have parametrized.

We consider null mean curvature flow along the standard lightcone in the de Sitter spacetime. This flow was first studied by Roesch–Scheuer along null hypersurfaces for the detection of MOTS, and independently by the author in the specific case of the standard Minkowski lightcone. Similar to the Minkowski case, null mean curvature flow along the de Sitter lightcone can be related to $2d$-Ricci flow for surfaces of genus $0$ by an appropriate rescaling. Building on this rescaling procedure, we analyse singularity formation, asymptotic behavior and ancient solutions to the flow.

We are concerned in this article with a classical topic in spectral geometry dating back to McKean-Singer, Patodi and Tanno: whether or not the constancy of sectional curvature (resp. holomorphic sectional curvature) of a compact Riemannian manifold (resp. Kähler manifold) can be completely determined by the eigenvalues of its $p$-Laplacian for a single integer $p$? We treat this question under two conditions: gradient shrinking Ricci soliton for Riemannian manifolds and cohomologically Einstein for Kähler manifolds. We show that, with some sporadic unknown cases, this is true for each $p$. Furthermore, we show that the condition of being isospectral can be relaxed to a suitable almost-isospectral version.

We show that any tangent cone of a singular shrinking Kähler-Ricci soliton is a normal affine algebraic variety. Moreover, the regular set of such a tangent cone in the metric sense coincides with the regular set in the algebraic sense. Along the way, we give a parabolic proof of Hörmander's $L^2$ estimate, which can be used to solve the $\overline{\partial }$-equation on any singular shrinking Kähler-Ricci soliton.

Glickenstein introduced the discrete conformal structures on polyhedral surfaces in an axiomatic approach from Riemannian geometry perspective. It includes Thurston's circle packings, Bowers-Stephenson's inversive distance circle packings and Luo's vertex scalings as special cases. In this paper, we study the deformation of Glickenstein's discrete conformal structures by combinatorial curvature flows. The combinatorial Ricci flow for Glickenstein's discrete conformal structures on triangulated surfaces is a generalization of Chow-Luo's combinatorial Ricci flow for Thurston's circle packings and Luo's combinatorial Yamabe flow for vertex scalings. We prove that the solution of the combinatorial Ricci flow for Glickenstein's discrete conformal structures on triangulated surfaces can be uniquely extended. Furthermore, under some necessary conditions, we prove that the solution of the extended combinatorial Ricci flow on a triangulated surface exists for all time and converges exponentially fast for any initial value. We further introduce the combinatorial Calabi flow for Glickenstein's discrete conformal structures on triangulated surfaces and study the basic properties of the flow. These combinatorial curvature flows provide effective algorithms for finding piecewise constant curvature metrics on surfaces with prescribed combinatorial curvatures.

Collapsed ancient solutions to the homogeneous Ricci flow on compact manifolds occur only on the total space of principal torus bundles. Under an algebraic assumption that guarantees flowing through diagonal metrics and a tameness assumption on the collapsing directions, we prove that such solutions have additional symmetries, i.e., they are invariant under the right action of their collapsing torus. As a byproduct of these additional torus symmetries, we prove that these solutions converge, backward in time, in the Gromov-Hausdorff topology to an Einstein metric on the base of a torus bundle.

We prove the following result: Let $(M,g_0)$ be a complete noncompact manifold of dimension $n\ge 12$ with isotropic curvature bounded below by a positive constant, with scalar curvature bounded above, and with injectivity radius bounded below. Then there is a finite collection $\mathcal{X}$ of spherical $n$-manifolds and manifolds of the form ${\mathbb{S}}^{n-1}\times \mathbb{R}/G$, where $G$ is a discrete subgroup of the isometry group of the round cylinder ${\mathbb{S}}^{n-1}\times \mathbb{R}$, such that $M$ is diffeomorphic to a (possible infinite) connected sum of members of $\mathcal{X}$. This extends a recent work of Huang. The proof uses Ricci flow with surgery on open orbifolds with isolated singularities.

In this article we examine the formation of cylindrical neckpinch singularities in Ricci flow in compact manifolds. Rigorous examples of neckpinch sigularity for rotationally symmetric initial data were first constructed by Angenent and Knopf, and examples with given asymptotic behaviour were constructed by them and Isenberg. Here we discuss the asymptotic behavior of the flow under Type-I assumption for general symmetric initial data, and show the previously constructed asymptotic profiles and are the only possibilities.

We give an algebraic criterion for a nilpotent real Lie algebra and prove that it provides a necessary and sufficient condition for the associated nilpotent Lie group to admit left-invariant Ricci solitons, called nilsolitons. As an application of this result, we generalise Nikolayevsky's criterion for the existence of nilsolitons to nilpotent Lie algebras without nice bases. We further prove a modified version of the Taketomi–Tamaru conjecture for nilpotent Lie groups which gives an obstruction to the existence of nilsolitons.

This paper investigates a generalized hyperbolic circle packing (including circles, horocycles or hypercycles) with respect to the total geodesic curvatures on the surface with boundary. We mainly focus on the existence and rigidity of circle packing whose contact graph is the $1$-skeleton of a finite polygonal cellular decomposition, which is analogous to the construction of Bobenko and Springborn [4]. Motivated by Colin de Verdière's method [6], we introduce the variational principle for generalized hyperbolic circle packings on polygons. By analyzing limit behaviours of generalized circle packings on polygons, we give an existence and rigidity for the generalized hyperbolic circle packing with conical singularities regarding the total geodesic curvature on each vertex of the contact graph. As a consequence, we introduce the combinatoral Ricci flow to find a desired circle packing with a prescribed total geodesic curvature on each vertex of the contact graph.

In this paper we study 4d gradient steady Ricci solitons, which are weak $\kappa$-solutions, and admit O(3)-symmetry. Under a weak curvature decay condition, we find precise geometric asymptotics of such solitons, which are similar to those for 3d compact $\kappa$-solutions found in [ABDS22]. This is the first step towards the classification of 4d gradient steady Ricci solitons and more general ancient Ricci flows.

In this paper, we prove that given a hyperbolic polyhedral metric with an inversive distance circle packing, and a target discrete curvature satisfying Gauss-Bonnet formula, there exist a unique inversive distance circle packing which is discrete conformal to the former one. We deform the surface by discrete Ricci flow, and do surgery by edge flipping when the orthogonal circles of some faces are about to be non-compact. The revised weighted Delaunay inequality of hyperbolic case implies the compactness of the orthogonal circle. We use a variational principle of a convex Ricci potential defined on the fiber bundles with cell-decomposition and differential structure based on Teichmüller space to finish the proof.

In this paper we consider $L^p$ Liouville type theorems for harmonic functions on gradient Ricci solitons. In particular, assume that $(M,g)$ is a gradient shrinking or steady Kähler-Ricci soliton, then we prove that any pluriharmonic function $u$ on $M$ with $\nabla u\in L^p(M)$ for some $1<p\le 2$ is a constant function.

Special Ricci-Hessian equations on Kähler manifolds $(M,g)$, as defined by Maschler [Ann. Global Anal. Geom. 34 (2008), 367–380] involve functions $\tau$ on $M$ and state that, for some function $\alpha$ of the real variable $\tau$, the sum of $\alpha \nabla d\tau$ and the Ricci tensor equals a functional multiple of the metric $g$, while $\alpha \nabla d\tau$ itself is assumed to be nonzero almost everywhere. Three well-known obvious "standard" cases are provided by (non-Einstein) gradient Kähler-Ricci solitons, conformally-Einstein Kähler metrics, and special Kähler-Ricci potentials. We show that, outside of these three cases, such an equation can only occur in complex dimension two and, at generic points, it must then represent one of three types, for which, up to normalizations, $\alpha =2\cot \tau$, or $\alpha =2\coth \tau$, or $\alpha =2\tanh \tau$. We also use the Cartan-Kähler theorem to prove that these three types are actually realized in a "nonstandard" way.

We establish a nonlinear analogue of a splitting map into a Euclidean space, as a harmonic map into a flat torus. We prove that the existence of such a map implies Gromov-Hausdorff closeness to a flat torus in any dimension. Furthermore, Gromov-Hausdorff closeness to a flat torus and an integral bound {on $r_M(x)$, the smallest eigenvalue of the Ricci tensor ${\text{ric}}_x$ in $x$}, imply the existence of a harmonic splitting map. Combining these results with Stern's inequality, we provide a new Gromov-Hausdorff stability theorem for flat $3$-tori. The main tools we employ include the harmonic map heat flow, Ricci flow, and both Ricci limits and RCD theories.

We prove that for any discrete curvature satisfying Gauss-Bonnet formula, there exist a unique up to scaling inversive distance circle packing in the discrete conformal equivalent class, whose polyhedral metric meets the target curvature. We prove it by constructing diffeomorphism between fiber bundles with cell decomposition based on Teichmüller spaces, and each discrete conformal equivalent class is a fiber passing through finite cell with respect to triangulations, which means we can do surgery on the discrete Ricci flow by edge flipping using a generalized Ptolemy equation to ensure it converge and never blow up.

We study the subsequential convergence of singular solutions to the Ricci flow with prescribed constant in space geodesic curvature on compact surfaces with boundary. Furthermore, we show that in the particular case of rotational symmetry, this convergence does not depend on the sign of the geodesic curvature of the boundary.

We develop a general theory of flows in the space of Riemannian metrics induced by neural network gradient descent. This is motivated in part by recent advances in approximating Calabi-Yau metrics with neural networks and is enabled by recent advances in understanding flows in the space of neural networks. We derive the corresponding metric flow equations, which are governed by a metric neural tangent kernel, a complicated, non-local object that evolves in time. However, many architectures admit an infinite-width limit in which the kernel becomes fixed and the dynamics simplify. Additional assumptions can induce locality in the flow, which allows for the realization of Perelman's formulation of Ricci flow that was used to resolve the 3d Poincaré conjecture. We demonstrate that such fixed kernel regimes lead to poor learning of numerical Calabi-Yau metrics, as is expected since the associated neural networks do not learn features. Conversely, we demonstrate that well-learned numerical metrics at finite-width exhibit an evolving metric-NTK, associated with feature learning. Our theory of neural network metric flows therefore explains why neural networks are better at learning Calabi-Yau metrics than fixed kernel methods, such as the Ricci flow.

We establish a structure result for the isometry group of non-compact, homogeneous manifolds admitting an immortal homogeneous Ricci flow solution.

In this paper, we consider the parabolic frequency for positive solutions of two nonlinear parabolic equations under the Ricci flow on closed manifolds. We obtain the monotonicity of parabolic frequency for the solution of two nonlinear parabolic equations with bounded Ricci curvature, then we apply the parabolic frequency monotonicity to get some integral type Harnack inequalities and we use -K1 instead of the lower bound 0 of Ricci curvature from Theorem 4.3 in 16, where K1 is any positive constant.

We consider noncollapsed steady gradient Ricci solitons with nonnegative sectional curvature. We show that such solitons always dimension reduce at infinity. This generalizes an earlier result in [CDM22] to higher dimensions. In dimension four, we classify possible reductions at infinity, which lays foundation for possible classifications of steady solitons. Moreover, we show that any tangent flow at infinity of a general noncollapsed steady soliton must split off a line. This generalizes an earlier result in [BCDMZ21] to higher dimensions. While this article is under preparation, we realized that part of our main results are proved independently in a recent post [ZZ23] under different assumptions.

The metric flow is introduced and extensively studied by Bamler [Bam20b, Bam20c], especially as an $\mathbb{F}$-limit of a sequence of smooth Ricci flows with uniformly bounded Nash entropy, in which case each regular point on the limit is a point of smooth convergence. In this note, we shall consider the $\mathbb{F}$-convergence of a sequence of $\mathbb{F}$-limit flows, and, like Bamler, show that each regular point on the limit is also a point of smooth convergence. The main result will be applied in a forthcoming work of the authors [CMZ23].

In the paper, we analysis the asymptotic behavior of noncompact $\kappa$-noncollapsed steady gradient Ricci soliton $(M,g)$ with nonnegative curvature operator away from a compact set $K$ of $M$. In particular, we prove: any $4d$ noncompact $\kappa$-noncollapsed steady gradient Ricci soliton $(M^4,g)$ with nonnegative sectional curvature must be a Bryant Ricci soliton up to scaling if it admits a sequence of rescaled flows of $(M^4,g)$, which converges subsequently to a family of shrinking quotient cylinders.

All known examples of simply-connected gradient Kähler-Ricci soliton in real dimension four are toric, and the symmetry is intrinsically related to the potential function $f$ and the scalar curvature $\text{SS}$. In this article, we consider the case that $f$ and $\text{SS}$ are functionally dependent and deduce a complete classification, while the independence case is addressed elsewhere. The main theorem recovers all known examples of cohomogeneity one symmetry. We also discover a connection to the theory of isoparametric functions and contact geometry. Indeed, a key ingredient is a new characterization for a deformed Sasakian structure generalizing a classical result.

This paper is devoted to the investigation of the monotonicity of parabolic frequency functional under conformal Ricci flow defined on a closed Riemannian manifold of constant scalar curvature and dimension not less than 3. Parabolic frequency functional for solutions of certain linear heat equation coupled with conformal pressure is defined and its monotonicity under the conformal Ricci flow is proved by applying Bakry-Emery Ricci curvature bounds. Some consequences of the monotonicity are also presented.

We establish geometric regularity for Type I blow-up limits of the Kähler-Ricci flow based at any sequence of Ricci vertices. As a consequence, the limiting flow is continuous in time in both Gromov-Hausdorff and Gromov-$W_1$ distance. In particular, the singular sets of each time slice and its tangent cones are close and of codimension no less than $4$.

We establish the scalar curvature and distance bounds, extending Perelman's work on the Fano Kähler-Ricci flow to general finite time solutions of the Kähler-Ricci flow. These bounds are achieved by our Li-Yau type and Harnack estimates for weighted Ricci potential functions of the Kähler-Ricci flow. We further prove that the Type I blow-ups of the finite time solution always sub-converge in Gromov-Hausdorff sense to an ancient solution on a family of analytic normal varieties with suitable choices of base points. As a consequence, the Type I diameter bound is proved for almost every fibre of collapsing solutions of the Kähler-Ricci flow on a Fano fibre bundle. We also apply our estimates to show that every solution of the Kähler-Ricci flow with Calabi symmetry must develop Type I singularities, including both cases of high codimensional contractions and fibre collapsing.

In this note, we give a new proof for Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds. The proof relies on a new Harnack estimate for a special family of functions in space-time. Our new approach initiates the work in for general finite time solutions of the Kähler-Ricci flow.

We give a new proof of the Smale conjecture for ${\mathbb{RP}}^3$ and all lens spaces using minimal surfaces and min-max theory. For ${\mathbb{RP}}^3$, the conjecture was first proved in 2019 by Bamler-Kleiner using Ricci flow.

In the present article, we characterize Clairaut conformal submersions whose total manifolds admit a Ricci soliton and provide a non-trivial example of such Clairaut conformal submersions. We firstly calculate scalar curvature and Ricci tensors of total manifolds of Clairaut conformal submersions and provide necessary conditions for the fibres of such Clairaut conformal submersions to be almost Ricci solitons and Einstein. Further, we provide necessary conditions for the base manifold to be Ricci soliton and Einstein. Then, we find a necessary condition for vector field $\dot{\zeta }$ to be conformal vector field and killing vector field. Besides, we indicate that if total manifolds of Clairaut conformal submersions admit a Ricci soliton with the potential mean curvature vector field $H$ of $Ker{\vartheta }_{\ast },$ then the total manifolds of Clairaut conformal submersions admit a gradient Ricci soliton. Finally, by solving Poisson equation, we acquire a necessary and sufficient condition for Clairaut conformal submersions to be harmonic.

The vacuum Robinson-Trautman solution admits a shear-free and twist-free null geodesic congruence with a nonvanishing expansion. We perform a comprehensive classification of solutions exhibiting this property in Einstein's gravity with a massless scalar field, assuming that the solution belongs at least to Petrov-type II and some of the components of Ricci tensor identically vanish. We find that these solutions can be grouped into three distinct classes: (I-a) a natural extension of the Robinson-Trautman family incorporating a scalar hair satisfying the time derivative of the Ricci flow equation, (I-b) a novel non-asymptotically flat solution characterized by two functions satisfying Perelman's pair of the Ricci flow equations, and (II) a dynamical solution possessing $\mathrm{SO}(3)$, $\mathrm{ISO}(2)$ or $\mathrm{SO}(1,2)$ symmetry. We provide a complete list of all explicit solutions falling into Petrov type D for classes (I-a) and (I-b). Moreover, leveraging the massless solution in class (I-a), we derive the neutral Robinson-Trautman solution to the $\mathcal{N}=2$ gauged supergravity with the prepotential $F(X)=-i{X}^0{X}^1$. By flipping the sign of the kinetic term of the scalar field, the Petrov-D class (I-a) solution leads to a time-dependent wormhole with an instantaneous spacetime singularity. Although the general solution is unavailable for class (II), we find a new dynamical solution with spherical symmetry from the AdS-Roberts solution via AdS/Ricci-flat correspondence.

By using the Ricci flow, we study local rigidity theorems regarding scalar curvature, isoperimetric constant and best constant of $L^2$ logarithmic Sobolev inequality. Precisely, we prove that if a metric $g$ on an open set $V$ in an $n$-dimensional Riemannian manifold satisfies $${\int }_{V}R(g)dvo{l}_g\ge 0\text{ and }I(V)\ge I({\mathbb{R}}^n),$$ or $${\int }_{V}R(g)dvo{l}_g\ge 0\text{ and }S(V)\ge S({\mathbb{R}}^n),$$ then $g=g_{{\mathbb{R}}^n}$ on $V$, where $R(g)$ is the scalar curvature of $g$, ${\mathbb{R}}^n$ is Euclidean space, $I(V)$ is the isoperimetric constant of $V$ and $S(V)$ is best constant of $L^2$ logarithmic Sobolev inequality of $V$. Moreover,we also obtain the local ${\mathbb{R}}^n$-rigidity about local Perelman's $\nu$-entropy, and local ${\mathbb{S}}^n$-rigidity (resp. ${\mathbb{H}}^n$-rigidity) theorems regarding the cases concerning $R(g)\ge n(n-1)$ (resp. $R(g)\ge -n(n-1)$), weighted isoperimetric constant and best constant of weighted $L^2$ logarithmic Sobolev inequality for the weighted metric ${\left(\cos \frac{d_g(p,x)}{2}\right)}^{-4}g$ (resp. ${\left(\cosh \frac{d_g(p,x)}{2}\right)}^{-4}g$).

We find Fano threefolds $X$ admitting Kähler-Ricci solitons (KRS) with non-trivial moduli, which are $\mathbb{T}$-varieties of complexity two. More precisely, we show that the weighted K-stability of $(X,{\xi }_0)$ (where ${\xi }_0$ is the soliton candidate) is equivalent to certain GIT-stability. In particular, this provides the first examples of strictly weighted K-semistable Fano varieties. On the other hand, we generalize Koiso's theorem to the log Fano setting. Indeed, we show that the K-stability of a log Fano pair $(V,{\Delta }_V)$ is equivalent to the weighted K-stability of a cone $(Y,{\Delta }_Y,{\xi }_0)$ over it. This also leads to new examples of KRS Fano varieties with non-trivial moduli and small automorphism groups. To achieve these, we establish the weighted Abban-Zhuang estimate generalizing the work of, which gives a lower bound of the weighted stability threshold ${\delta }_{\mathbb{T}}^g(X,\Delta )$. This is an effective way to check the weighted K-semistablity of a log Fano triple $(X,\Delta ,{\xi }_0)$. This estimate is also useful in testing (weighted) K-polystability based on the work of.

The classical Minkowski inequality implies that the volume of a bounded convex domain is controlled from above by the integral of the mean curvature of its boundary. In this note, we establish an analogous inequality without the convexity assumption for all bounded smooth domains in a complete manifold with its bottom spectrum being suitably large relative to its Ricci curvature lower bound. An immediate implication is the nonexistence of embedded compact minimal hypersurfaces in such manifolds. This nonexistence issue is also considered for steady and expanding Ricci solitons.

The aim of this paper is to study almost rigidity properties of super Ricci flow whose Muller quantity is non-negative. We conclude almost splitting and quantitative stratification theorems that have been established by Bamler for Ricci flow. As a byproduct, we obtain an almost constancy for a certain integral quantity concerning scalar curvature at an almost selfsimilar point, which is new even for Ricci flow.

Motivated by Guo-Luo's generalized circle packings on surfaces with boundary, we introduce the generalized sphere packings on 3-dimensional manifolds with boundary. Then we investigate the rigidity of the generalized sphere packing metrics. We prove that the generalized sphere packing metric is determined by the combinatorial scalar curvature. To find the hyper-ideal polyhedral metrics on 3-dimensional manifolds with prescribed combinatorial scalar curvature, we introduce the combinatorial Ricci flow and combinatorial Calabi flow for the generalized sphere packings on 3-dimensional manifolds with boundary. Then we study the longtime existence and convergence for the solutions of these combinatorial curvature flows.

We survey several problems concerning Riemannian manifolds with positive curvature of one form or another. We describe the PIC1 notion of positive curvature and argue that it is often the sharp notion of positive curvature to consider. Finally we explain how recent Ricci flow theory is particularly well adapted to solve these problems.

We provide a derivative estimate for the pluriclosed flow, controlling higher order derivatives of Chern curvature and torsion using the Chern curvature. Moreover, we derive an estimate for torsion tensor using Chern Ricci curvature in dimension two. And in the Hermitian-symplectic case, we find a monotonic quantity and use it to prove that all Hermitian-symplectic solitons are Kähler Ricci solitons.

In this paper, we show that the singularity type of solutions to the Käher-Ricci flow on a numerically effective manifold does not depend on the initial metric. More precisely if there exists a type III solution to the Kähler-Ricci flow, then any other solution starting from a different initial metric will also be Type III. This generalises previous results by Y. Zhang for the semi-ample case.

We prove the existence and uniqueness of a weighted analogue of the Fefferman-Graham ambient metric for manifolds with density. We then show that this ambient metric forms the natural geometric framework for the singular Ricci flow: given a singular gradient Ricci flow spacetime in the Kleiner-Lott sense, we construct a unique global ambient half-space from it. We also prove the converse, that every global ambient space contains a singular gradient Ricci flow spacetime, thereby completing the correspondence. Our main application is the construction of infinite families of fully non-linear analogues of Perelman's $\mathcal{F}$ and $\mathcal{W}$ functionals. We extend Perelman's monotonicity result to these two families of functionals under several conditions, including for shrinking solitons and Einstein manifolds. We do so by constructing a "Ricci flow vector field" in the ambient space, which may be of independent research interest. We also prove that the weighted GJMS operators associated with the weighted ambient metric are formally self-adjoint, and that the associated weighted renormalized volume coefficients are variational.

We introduce a classification conjecture for $\kappa$-solutions in 4d Ricci flow. Our conjectured list includes known examples from the literature, but also a new 1-parameter family of ${\mathbb{Z}}_2^2\times {\mathrm{O}}_3$-symmetric bubble-sheet ovals that we construct. We observe that some special cases of the conjecture follow from recent results in the literature. We also introduce a stronger variant of the classification conjecture for ancient asymptotically cylindrical 4d Ricci flows, which does not assume smoothness and nonnegative curvature operator a priori. Assuming this stronger variant holds true, we establish a canonical neighborhood theorem for 4d Ricci flow through cylindrical singularities, which shares some elements in common with Perelman's canonical neighborhood theorem for 3d Ricci flow as well as the mean-convex neighborhood theorem for mean curvature flow through neck-singularities. Finally, we argue that quotient-necks lead to new phenomena, and sketch an example of non-uniqueness for 4d Ricci flow through singularities.

The investigation of Ricci solitons is the focus of this work. We have proved triviality results for compact gradient Ricci soliton under certain restriction. Later, a rigidity result is derived for a compact gradient shrinking Ricci soliton. Also, we have estimated the conjugate radius for non-compact gradient shrinking Ricci solitons with superharmonic potential. Moreover, an upper bound for the conjugate radius of Ricci soliton with concircular potential vector field is determined. Finally, it is proved that a non-compact gradient Ricci soliton with a pole and non-negative Ricci curvature is non-shrinking.

We prove that there exists a gradient expanding Ricci soliton asymptotic to any given cone over the product of a round sphere and a Ricci flat manifold. In particular we obtain asymptotically conical expanding Ricci solitons with positive scalar curvature on ${\mathbb{R}}^3\times S^1.$ More generally we construct continuous families of gradient expanding Ricci solitons on trivial vector bundles over products of Einstein manifolds with arbitrary Einstein constants.

We make classifications of gradient Ricci solitons $(M,g,f)$ with harmonic Weyl curvature. As a local classification, we prove that the soliton metric $g$ is locally isometric to one of the following four types: an Einstein manifold, the Riemannian product of a Ricci flat manifold and an Einstein manifold, a warped product of $\mathbb{R}$ and an Einstein manifold, and a singular warped product of ${\mathbb{R}}^2$ and a Ricci flat manifold. Compared with the previous four-dimensional study in, we have developed a novel method of {\it refined adapted frame fields} and overcome the main difficulty arising from a large number of Riemmannian connection components in dimension$\ge 5$. Next we have obtained a classification of {\it complete} gradient Ricci solitons with harmonic Weyl curvature. For the proof, using the real analytic nature of $g$ and $f$, we elaborate geometric arguments to fit together local regions.

We introduce a large class of canonical Kähler metrics, called in this paper well-behaved, extending metrics induced by complex space forms. We study Kähler–Ricci iterations of well-behaved metrics on compact and non-compact Kähler manifolds. That is, we are interested in well-behaved metrics for which the iteration of the Ricci operator is a multiple of a Kähler metric, i.e., ${\rho }_{\omega }^k=\lambda \Omega$. In particular, when $k=1$, under some condition on the maximal domain of definition of canonical coordinates, we show that $\lambda$ is forced to be positive. Moreover, for arbitrary $k$, we prove two additional results. Namely, if $\omega$ and $\Omega$ are induced by a flat metric, then $\omega$ is Ricci-flat. Finally, if a Kähler-Ricci soliton $\Omega$ arises as Kähler–Ricci iteration of a metric $\omega$ induced by a complex space form, then the Kähler–Ricci soliton is forced to be trivial, that is, Kähler–Einstein. These three theorems extend well known results on Kähler–Einstein metrics to higher iterations of the Ricci operator and a larger class of metrics.

We prove matrix Li-Yau-Hamilton estimates for positive solutions to the heat equation and the backward conjugate heat equation, both coupled with the Kähler-Ricci flow. As an application, we obtain a monotonicity formula.

In order to reconcile the wave-function collapse in quantum mechanics with the finiteness of signals' propagation in general relativity, we delve into a stochastic version of the Ricci flow and study its non-relativistic limit in presence of matter. We hence derive the Diósi-Penrose collapse model for the wave-function of a quantum gas. The procedure entails additional parameters with respect to phenomenological models hitherto accounted for, including the temperature of the gas and the cosmological constant, in turn related to the stochastic gravitational noise responsible for the collapse.

After an introductory chapter on the quantum supersymmetric string, in which particular attention will be devoted to the techniques via which phenomenologically viable models can be obtained from the ultraviolet microscopic degrees of freedom, and a brief review of the swampland program, the technical tools required to deal with geometric flows will be outlined. The evolution of a broad family of scalar and metric bubble solutions under Perelman's combined flow will be then discussed, together with their asymptotic behaviour. Thereafter, the geometric flow equations associated to a generalised version of Perelman's entropy function will be derived and employed in defining the action-induced flow associated to a given theory for a scalar field and a dynamical metric. The problem of preserving Einstein field equations along the corresponding moduli space trajectories will be cured by allowing a supplementary energy-momentum tensor term to appear along the flow. In a particular example, such contribution will be shown to precisely reproduce the infinite tower of states with exponentially dropping masses postulated by the distance conjecture.

Lai (2021) used singular Ricci flows, introduced by Kleiner and Lott (2017), to construct a nonnegative Ricci curvature Ricci flow $g(t)$ emerging from an arbitrary 3D complete noncompact Riemannian manifold $(M^3,g_0)$ which has nonnegative Ricci curvature. We show $g(t)$ is complete for positive times provided $g_0$ satisfies a volume ratio lower bound that approaches zero at spatial infinity. Our proof combines a pseudolocality result of Lai (2021) for singular flows, together with a pseudolocality result of Hochard (2016) and Simon and Topping (2022) for nonsingular flows. We also show that the construction of complete nonnegative complex sectional curvature flows by Cabezas-Rivas and Wilking (2015) can be adapted here to show $g(t)$ is complete for positive times provided $g_0$ is a compactly supported perturbation of a nonnegative sectional curvature metric on ${\mathbb{R}}^3$.

The family of invariant Riemannian manifolds in the Wallach flag manifold $\mathrm{SU}(3)/{\mathrm{T}}^2$ is described by three parameters $(x,y,z)$ of positive real numbers. By restricting such a family of metrics in the tetrahedron $\mathcal{T}:=x+y+z=1$, in this paper, we describe all regions $\mathcal{R}\subset \mathcal{T}$ admitting metrics with curvature properties varying from positive sectional curvature to positive scalar curvature, including positive intermediate curvature notion's. We study the dynamics of such regions under the projected Ricci flow in the plane $(x,y)$, concluding sign curvature maintenance and escaping. In addition, we obtain some results for positive intermediate Ricci curvature for a path of metrics on fiber bundles over $\mathrm{SU}(3)/{\mathrm{T}}^2$, further studying its evolution under the Ricci flow on the base.

We define a geometric quantity for asymptotically hyperbolic manifolds, which we call the volume-renormalized mass. It is essentially a linear combination of the ADM mass surface integral and a renormalization of the volume. We show that the volume-renormalized mass is well-defined and diffeomorphism invariant under weaker fall-off conditions than required to ensure that the renormalized volume and the ADM mass surface integral are well-defined separately. We prove several positivity results for the volume-renormalized mass. We also use it to define a renormalized Einstein–Hilbert action and a renormalized expander entropy which is nondecreasing under the Ricci flow. Further, we show that local maximizers of the entropy are local minimizers of the volume-renormalized mass.

It is shown that on compact $Spin(7)$–manifold with exterior derivative of the Lee form lying in the Lie algebra $spin(7)$ the curvature $R$ of the $Spin(7)$–torsion connection $R\in S^2{\Lambda }^2$ with vanishing Ricci tensor if and only if the $3$-form torsion is parallel with respect to the Levi-Civita connection. It is also proved that $R$ satisfies the Riemannian first Bianchi identity exactly when the $3$-form torsion is parallel with respect to the Levi-Civita and to the $Spin(7)$–torsion connections simultaneously. Precise conditions for a compact $Spin(7)$–manifold to has closed torsion are given in terms of the Ricci tensor of the $Spin(7)$–torsion connection. It is shown that a compact $Spin(7)$–manifold with closed torsion is Ricci flat if and only if either the norm of the torsion or the Riemannian scalar curvature is constant. It is proved that any compact $Spin(7)$–manifold with closed torsion 3-form is a generalized gradient Ricci soliton and this is equivalent to a certain vector field to be parallel with respect to the torsion connection. In particular, this vector field preserves the $Spin(7)$–structure.

Curvature properties of the characteristic connection on an integrable $G_2$ manifold are investigated. We consider integrable $G_2$ manifold of constant type, i.e. the scalar product of the exterior derivative of the $G_2$ form with its Hodge dual is a constant. We show that on an integrable $G_2$ manifold of constant type with $G_2$-instanton characteristic curvature and vanishing Ricci tensor the torsion 3-form is harmonic. Consequently, we prove that the characteristic curvature is symmetric in exchange the first and the second pair and Ricci flat if and only if the three-form torsion is parallel with respect to the Levi-Civita and to the characteristic connection simultaneously and this is equivalent to the condition that the characteristic curvature satisfies the Riemannian first Bianchi identity. We find that the Hull connection is a $G_2$-instanton exactly when the torsion is closed. We observe that any compact integrable $G_2$ manifold with closed torsion is a generalized gradient Ricci soliton and this is equivalent to a certain vector field to be parallel with respect to the characteristic connection. In particular, this vector field is an infinitesimal automorphism of the $G_2$ structure and preserves the torsion three form.

The second author and H. Yin have developed a Ricci flow existence theory that gives a complete Ricci flow starting with a surface equipped with a conformal structure and a nonatomic Radon measure as a volume measure. This led to the discovery of a large array of new expanding Ricci solitons. In this paper we use the recent uniqueness theory in this context, also developed by the second author and H. Yin, to give a complete classification of all expanding Ricci solitons on surfaces. Along the way, we prove a converse to the existence theory that is not constrained to solitons: every complete Ricci flow on a surface over a time interval $(0,\epsilon )$ admits a $t\downarrow 0$ limit within the class of admissible initial data. This makes surfaces the first nontrivial setting for Ricci flow in which a bijection can be given between the entire set of complete Ricci flows over maximal time intervals $(0,T)$, and a class of initial data that induces them.

It is observed that on a compact almost complex Calabi-Yau with torsion 6-manifold the Nijenhuis tensor is parallel with respect to the torsion connection. If the torsion is closed then the space is a compact generalized gradient Ricci soliton. In this case, the torsion connection is Ricci-flat if and only if either the norm of the torsion or the Riemannian scalar curvature is constant. On a compact almost complex Calabi-Yau with torsion 6-manifold it is shown that the curvature of the torsion connection is symmetric on exchange of the first and the second pairs and has vanishing Ricci tensor if and only if it satisfies the Riemannian first Bianchi identity.

Curvature properties of a metric connection with totally skew-symmetric torsion are investigated. It is shown that if either the 3-form $T$ is harmonic, $dT=\delta T=0$ or the curvature of the torsion connection $R\in S^2{\Lambda }^2$ then the scalar curvature of a $\nabla$-Einstein manifold is determined by the norm of the torsion up to a constant. It is proved that a compact generalized gradient Ricci soliton with closed torsion is Ricci flat if and only if either the norm of the torsion or the Riemannian scalar curvature are constants. In this case the torsion 3-form is harmonic and the gradient function has to be constant. Necessary and sufficient conditions a metric connection with skew torsion to satisfy the Riemannian first Bianchi identity as well as the contracted Riemannian second Binachi identity are presented. It is shown that if the torsion connection satisfies the Riemannian first Bianchi identity then it satisfies the contracted Riemannian second Bianchi identity. It is also proved that a metric connection with skew torsion satisfying the curvature identity $R(X,Y,Z,V)=R(Z,Y,X,V)$ must be flat.

Recently, Qi S.Zhang [26] has derived a sharp Li-Yau estimate for positive solutions of the heat equation on closed Riemannian manifolds with the Ricci curvature bounded below by a negative constant. The proof is based on an integral iteration argument which utilizes Hamilton's gradient estimate, heat kernel Gaussian bounds and parabolic Harnack inequality. In this paper, we show that the sharp Li-Yau estimate can actually be obtained directly following the classical maximum principle argument, which simplifies the proof in [26]. In addition, we apply the same idea to the heat and conjugate heat equations under the Ricci flow and prove some Li-Yau type estimates with optimal coefficients.

The Bach tensor is classically defined in dimension 4, and work from J. Bergman and others shows that $B=\frac{1}{2}U+\frac{1}{6}V$ where $U$ and $V$ are more basic 2-tensors, which are symmetric, divergence-free, algebraically independent, and quadratic in the Riemann tensor. In this paper, we extend H.-D. Cao and Q. Chen's results for Bach-flat gradient shrinking Ricci solitons to solitons with $\mathfrak{B}=\alpha U+\beta V=0$.

We discuss local Sasakian immersion of Sasaki-Ricci solitons (SRS) into fiber products of homogeneous Sasakian manifolds. In particular, we prove that SRS locally induced by a large class of fiber products of homogeneous Sasakian manifolds are, in fact, $\eta$-Einstein. The results are stronger for immersions into Sasakian space forms. Moreover, we show an example of a Kähler-Ricci soliton on ${\mathbb{C}}^n$ which admits no local holomorphic isometry into products of homogeneous bounded domains with flat Kähler manifolds and generalized flag manifolds.

Following the recent development by Guo-Phong-Tong and Chen-Cheng, we derived the $L^{\infty }$ estimate for Kähler-Ricci flows under a weaker assumption. The technique also extends to more general cases coming from different geometric backgrounds.

The stability of a recently developed piecewise flat Ricci flow is investigated, using a linear stability analysis and numerical simulations, and a class of piecewise flat approximations of smooth manifolds is adapted to avoid an inherent numerical instability. These adaptations have also been used in a related paper to show the convergence of the piecewise flat Ricci flow to known smooth Ricci flow solutions for a variety of manifolds.

In this paper we prove matrix Li-Yau-Hamilton estimates for positive solutions to the heat equation and the backward conjugate heat equation, both coupled with the Ricci flow. We then apply such estimates to establish the monotonicity of parabolic frequencies up to correction factors. As applications, we obtain some unique continuation results under the nonnegativity of sectional or complex sectional curvature.