A Modified Kähler-Ricci Flow

Pluripotential Monge-Amp{è}re flows in Big Cohomology Classes

We study pluripotential complex Monge-Ampère flows in big cohomology classes on compact K{ä}hler manifolds. We use the Perron method, considering pluripotential subsolutions to the Cauchy problem. We prove that, under natural assumptions on the data, the upper envelope of all subsolutions is continuous in space and semi-concave in time, and provides a unique pluripotential solution with such regularity. We apply this theory to study pluripotential K{ä}hler-Ricci flows on compact K{ä}hler manifolds of general type as well as on stable varieties.

Ricci flow on modified Riemann extensions

We study the properties of Modified Riemann extensions evolving under Ricci flow. We obtain the necessary and sufficient condition for modified Riemann extension under Ricci flow to stay as modified Riemann extension. We also discuss the properties of the curvature tensors under Ricci flow.

Smooth approximation of the modified conical Kähler-Ricci flow

We introduce the conical Kähler-Ricci flow modified by a holomorphic vector field. We construct a long-time solution of the modified conical Kähler-Ricci flow as the limit of a sequence of smooth Kähler-Ricci flows.

Modified Kähler-Ricci flow on projective bundles

In this paper, we propose a method of studying the modified Kahler-Ricci flow on projective bundles and give the explicit equation from the view point of symplectic geometry.

The modified Kähler-Ricci flow and solitons

We investigate the Kähler-Ricci flow modified by a holomorphic vector field. We find equivalent analytic criteria for the convergence of the flow to a Kähler-Ricci soliton. In addition, we relate the asymptotic behavior of the scalar curvature along the flow to the lower boundedness of the modified Mabuchi energy.

Lower bound of modified $K$-energy on a Fano manifold with degeneration for Kähler-Ricci solitons

In this paper, we extend Tosatti's method to study the lower boundedness of modified $K$-energy on a Fano manifold and apply this result to study the relative $K$-stability of the deformation space of a Kähler Ricci soliton.

拟爱因斯坦度量的刻画

A Characterization of Quasi-Einstein Metrics

本文研究修正的Ricci孤子，作为包含Ricci孤子和$n$-拟爱因斯坦度量的新类，并证明常数量曲率紧流形上梯度修正Ricci几乎孤子标准球面的刚性，给出修正Ricci-调和孤子例子。

梯度收缩Ricci孤子与修正截面曲率

Gradient Shrinking Ricci Solitons and Modified Sectional Curvature

研究具有正修正截面曲率的四维梯度收缩Ricci孤子，通过特定条件证明其局部为Kähler，得到紧致孤子的特征定理等结果，并建立Hitchin - Thorpe型不等式，为流形提供拓扑约束。

Laplacian comparison theorem on Riemannian manifolds with modified m-Bakry-Emery Ricci lower bounds for $m\leq1$

In this paper, we prove a Laplacian comparison theorem for non-symmetric diffusion operator on complete smooth $n$-dimensional Riemannian manifold having a lower bound of modified $m$-Bakry-Émery Ricci tensor under $m\leq 1$ in terms of vector fields. As consequences, we give the optimal conditions for modified $m$-Bakry-Émery Ricci tensor under $m\leq1$ such that the (weighted) Myers' theorem, Bishop-Gromov volume comparison theorem, Ambrose-Myers' theorem, Cheng's maximal diameter theorem, and the Cheeger-Gromoll type splitting theorem hold. Some of these results were well-studied for $m$-Ba…

Asymptotic stability for Kähler-Ricci solitons

We show that the coercivity of the modified Ding functional leads to the existence of a certain kind of balanced metrics and their convergence to the Kähler-Ricci soliton modulo automorphisms. In our results, we do not assume that the vanishing of the higher order modified Futaki invariants introduced by Berman-Nyström.

具有从下方有界的修正Ricci曲率的完备流形上Witten拉普拉斯算子和加权p-拉普拉斯算子的特征值比较定理

Eigenvalue comparison theorems for the Witten-Laplacian and the weighted $p$-Laplacian on complete manifolds with a modified Ricci curvature bounded from below

在具有从下方有界的修正Ricci曲率的完备流形上，建立Witten拉普拉斯算子和加权p-拉普拉斯算子在测地球上的第一狄利克雷特征值的程氏型特征值比较定理。

Structure of collapsing solutions of generalized Ricci flow

We derive modified Perelman-type monotonicity formulas for solutions to the generalized Ricci flow equation with symmetry on principal bundles, which lead to rigidity and classification results for nonsingular solutions.

A second variation formula for Perelman's $\mathcal{W}$-functional along the modified Kähler-Ricci flow

We show a quite simple second variation formula for Perelman's $\mathcal{W}$-functional along the modified Kähler-Ricci flow over Fano manifolds.

修正Ricci流的特征值下界与分裂定理

Eigenvalue lower bounds and splitting for modified Ricci flow

该研究针对修正Ricci流，证明了滑移拉普拉斯算子特征值的精确下界，并给出了等号成立时的分裂定理，丰富了Ricci流相关理论。

On the Convergence of a Modified Kaehler-Ricci flow

We study the convergence of a modified Kaeher-Ricci flow defined by Zhou Zhang. We show that the flow converges to a singular metric when the limit class is degenerate. This proves a conjecture of Zhang.

Bartnik's mass and Hamilton's Modified Ricci Flow

We provide estimates on the Bartnik mass of constant mean curvature (CMC) surfaces which are diffeomorphic to spheres and have positive mean curvature. We prove that the Bartnik mass is bounded from above by the Hawking mass and a new notion we call the asphericity mass. The asphericity mass is defined by applying Hamilton's modified Ricci flow and depends only upon the restricted metric of the surface and not on its mean curvature. The theorem is proven by studying a class of asymptotically flat Riemannian manifolds foliated by surfaces satisfying Hamilton's modified Ricci flow with prescribe…

Fano manifolds with weak almost Kähler-Ricci solitons

In this paper, we prove that a sequence of weak almost Kähler-Ricci solitons under further suitable conditions converge to a Kähler-Ricci soliton with complex codimension of singularities at least 2 in the Gromov-Hausdorff topology. As a corollary, we show that on a Fano manifold with the modified K-energy bounded below, there exists a sequence of weak almost Kähler-Ricci solitons which converge to a Kähler-Ricci soliton with complex codimension of singularities at least 2 in the Gromov-Hausdorff topology.

The modified Kähler-Ricci flow, II

We improve the understanding of both finite time and infinite time singularities of the modified Kähler-Ricci flow as initiated by the second author of this paper in [26]. This is done by relating the modified Kähler-Ricci flow with the recent studies on the classic Kähler-Ricci flow and the degenerate complex Monge-Ampère equation.

Twisted and conical Kähler-Ricci soliton on Fano manifolds

In this paper, we consider the twisted Kähler-Ricci soliton, and show that the existence of twisted Kähler-Ricci soliton with semi-positive twisting form is closely related to the properness of some energy functionals. We also consider the conical Kähler-Ricci soliton, and obtain some existence results. In particular, under some assumptions on the divisor and $α$-invariant, we get the properness of the modified log K-energy and the existence of conical Kähler-Ricci soliton with suitable cone angle.

Mean curvature flow in a Ricci flow background

Following work of Ecker, we consider a weighted Gibbons-Hawking-York functional on a Riemannian manifold-with-boundary. We compute its variational properties and its time derivative under Perelman's modified Ricci flow. The answer has a boundary term which involves an extension of Hamilton's Harnack expression for the mean curvature flow in Euclidean space. We also derive the evolution equations for the second fundamental form and the mean curvature, under a mean curvature flow in a Ricci flow background. In the case of a gradient Ricci soliton background, we discuss mean curvature solitons an…

Supremum of Perelman's entropy and Kähler-Ricci flow on a Fano manifold

In this paper, we extend the method in [TZhu5] to study the energy level $L(\cdot)$ of Perelman's entropy $λ(\cdot)$ for Kähler-Ricci flow on a Fano manifold. Consequently, we first compute the supremum of $λ(\cdot)$ in Kähler class $2πc_1(M)$ under an assumption that the modified Mabuchi's K-energy $μ(\cdot)$ defined in [TZhu2] is bounded from below. Secondly, we give an alternative proof to the main theorem about the convergence of Kähler-Ricci flow in [TZhu3].

Stability of the Ricci Yang-Mills flow at Einstein Yang-Mills metrics

Let $P$ be a principal U(1)-bundle over a closed manifold $M$. On $P$, one can define a modified version of the Ricci flow called the Ricci Yang-Mills flow, due to these equations being a coupling of Ricci flow and the Yang-Mills heat flow. We use maximal regularity theory and ideas of Simonett concerning the asymptotic behavior of abstract quasilinear parabolic partial differential equations to study the stability of the volume-normalized Ricci Yang-Mills flow at Einstein Yang-Mills metrics in dimension two. In certain cases, we show the presence of a center manifold of fixed points, while in…

伪广义Ricci递推时空与修正引力

Pseudo generalized Ricci-recurrent spacetimes and modified gravity

该研究引入并刻画伪广义Ricci递推时空并验证其存在性，证明其相关共形平坦情形为伪拟Einstein时空、其广义Robertson-Walker形式为理想流体时空，还推导了其在两类修正引力下的能量条件。

伪广义Ricci递推时空及其在修正引力中的若干应用

Pseudo generalized Ricci-recurrent spacetimes with certain applications to modified gravity

本文引入伪广义Ricci递推时空并证明其存在，分析了其在广义Robertson-Walker时空与完美流体时空中的性质，还研究了其在$f(\mathcal{R})$引力下的能量条件。

A steady length function for Ricci flow

A fundamental step in the analysis of singularities of Ricci flow was the discovery by Perelman of a monotonic volume quantity which detected shrinking solitons in (arXiv:math/0211159). A similar quantity was found by Feldman, Ilmanen, and Ni in 2005 which detected expanding solitons. The current work introduces a modified length functional as a first step towards a steady soliton monotonicity formula. This length functional generates a distance function in the usual way which is shown to satisfy several differential inequalities which saturate precisely on manifolds satisfying a modification …

The rate of $\mathbb{F}$-convergence for Ricci flows with closed and smooth tangent flows

This article is a continuation of [CMZ21b], where we proved that a Ricci flow with a closed and smooth tangent flow has unique tangent flow, and its corresponding forward or backward modified Ricci flow converges in the rate of $t^{-β}$ for some $β>0$. In this article, we calculate the corresponding $\mathbb{F}$-convergence rate: after being scaled by a factor $λ>0$, a Ricci flow with closed and smooth tangent flow is $|\log λ|^{-θ}$ close to its tangent flow in the $\mathbb{F}$-sense, where $θ$ is a positive number, $λ\gg 1$ in the blow-up case, and $λ\ll 1$ in the blow-down case.

Kähler-Ricci flow for deformed complex structures

Let $(M,J_0)$ be a Fano manifold which admits a Kähler-Ricci soliton, we analyze the behavior of the Kähler-Ricci flow near this soliton as we deform the complex structure $J_0$. First, we will establish an inequality of Lojasiewicz's type for Perelman's entropy along the Kähler-Ricci flow. Then we prove the convergence of Kähler-Ricci flow when the complex structure associated to the initial value lies in the kernel $Z$ or negative part of the second variation operator of Perelman's entropy. As applications, we solve the Yau-Tian-Donaldson conjecture for the existence of Kähler-Ricci solitons…

Stability of Kähler-Ricci flow in the space of Kähler metrics

In this paper, we prove that on a Fano manifold $M$ which admits a Kähler-Ricci soliton $(\om,X)$, if the initial Kähler metric $\om_{\vphi_0}$ is close to $\om$ in some weak sense, then the weak Kähler-Ricci flow exists globally and converges in Cheeger-Gromov sense. Moreover, if $\vphi_0$ is also $K_X$-invariant, then the weak modified Kähler-Ricci flow converges exponentially to a unique Kähler-Ricci soliton nearby. Especially, if the Futaki invariant vanishes, we may delete the $K_X$-invariant assumption. The methods based on the metric geometry of the space of the Kähler metrics are poten…

Ricci流中柱状奇点处切流的强唯一性

Strong uniqueness of tangent flows at cylindrical singularities in Ricci flow

本文在基点附近几何接近标准柱面或其商结构的假设下，建立了一个指向$\mathcal{W}$-熵的Lojasiewicz不等式，并证明了Ricci流首次奇异时间处柱状切流的强唯一性。

可变耦合参数下扩展 Ricci 流中几何常数的演化与单调性

Evolution and Monotonicity of Geometric Constants under Extended Ricci Flows with Variable Coupling Parameters

研究在含可变耦合参数的扩展 Ricci 流下几何常数 λ 的演化，通过修改演化方程系数推导其在扩展与归一化流下的公式，并给出单调性成立条件。

Measure Estimates, Harnack Inequalities and Ricci Lower Bound

On a Riemannian metric-measure space, we establish an Alexandrov-Bakelman-Pucci type measure estimate connecting Bakry-Émery Ricci curvature lower bound, modified Laplacian and the measure of certain special sets. We apply this estimate to prove Harnack inequalities for the modified Laplacian operator and fully non-linear operators. These inequalities seem not available in the literature; And our proof, solely based on the ABP estimate, does not involve any Sobolev inequalities nor gradient estimate. We also propose a question regarding the characterization of Ricci lower bound by the Harnack …

On the modified Futaki invariant of complete intersections in projective spaces

It is known that a necessary condition for the existence of Kähler-Ricci solitons is the vanishing of the modified Futaki invariant introduced by Tian-Zhu. In a recent work of Berman-Nyström, it was generalized for (singular) Fano varieties and the notion of algebro-geometric stability of the pair $(M,V)$ of a Fano manifold $M$ and a holomorphic vector field $V$ was introduced. In this paper, we propose a method of computing the modified Futaki invariant for Fano complete intersections in projective spaces.

The Soliton Kähler-Ricci Flow over Fano Manifolds

We introduce a flow of Kähler structures over Fano manifolds with formal limit at infinite time a Kähler-Ricci soliton. This flow correspond to a Perelman's modified backward Kähler-Ricci type flow that we call Soliton-Kähler-Ricci flow. It can be generated by the Soliton-Ricci flow. We assume that the Soliton-Ricci flow exists for all times and the Bakry-Emery-Ricci tensor preserve a positive uniform lower bound with respect to the evolving metric. In this case we show that the corresponding Soliton-Kähler-Ricci flow converges exponentially fast to a Kähler-Ricci soliton.

Space-time constructions for the mean curvature flow in a Ricci flow background

Given a solution of the (backwards) Ricci flow one can construct a so called canonical soliton metric on space-time, introduced by E. Cabezas-Rivas and P. Topping. We observe that for a mean curvature flow within a (backwards) Ricci flow background, the space-time track of the mean curvature flow yields a canonical soliton of the coupled flow within the canonical Ricci soliton. We show that this provides a link between Hamilton's differential Harnack estimate for the mean curvature flow and Hamilton's differential Harnack estimate for the Ricci flow. Moreover the second fundamental form of our…

The Soliton-Ricci Flow over Compact Manifolds

We introduce a flow of Riemannian metrics over compact manifolds with formal limit at infinite time a shrinking Ricci soliton. We call this flow the Soliton-Ricci flow. It correspond to a Perelman's modified backward Ricci type flow with some special restriction conditions. The restriction conditions are motivated by convexity results for Perelman's $\mathcal{W}$-functional over convex subsets inside adequate subspaces of Riemannian metrics. We show indeed that the Soliton-Ricci flow is generated by the gradient flow of the restriction of Perelman's $\mathcal{W}$-functional over such subspaces…

$\cF$-functional and geodesic stability

We consider canonical metrics on Fano manifolds. First we introduce a norm-type functional on Fano manifolds, which has Kahler-Einstein or Kahler-Ricci soliton as its critical point and the Kahler-Ricci flow can be viewed as its (reduced) gradient flow. We then obtain a natural lower bound of this functional. As an application, we prove that Kahler-Ricci soliton, if exists, maximizes Perelman's $μ$-functional without extra assumptions. Second we consider a conjecture proposed by S.K. Donaldson in terms of $\cK$-energy. Our simple observation is that $\cF$-functional, as $\cK$-energy, also inte…

The Ricci Flow on Domains in Cohomogeneity One Manifolds

Suppose $G$ is a compact Lie group, $H$ is a closed subgroup of $G$, and the homogeneous space $G/H$ is connected. The paper investigates the Ricci flow on a manifold $M$ diffeomorphic to $[0,1]\times G/H$. First, we prove a short-time existence and uniqueness theorem for a $G$-invariant solution $g(t)$ satisfying the boundary condition $\mathop{\mathrm{II}}(g(t))=F(t,g_{\partial M}(t))$ and the initial condition $g(0)=\hat g$. Here, $\mathop{\mathrm{II}}(g(t))$ is the second fundamental form of $\partial M$, $g_{\partial M}$ is the metric induced on $\partial M$ by $g(t)$, $F$ is a smooth map…

Using 3D Stringy Gravity to Understand the Thurston Conjecture

We present a string inspired 3D Euclidean field theory as the starting point for a modified Ricci flow analysis of the Thurston conjecture. In addition to the metric, the theory contains a dilaton, an antisymmetric tensor field and a Maxwell-Chern Simons field. For constant dilaton, the theory appears to obey a Birkhoff theorem which allows only nine possible classes of solutions, depending on the signs of the parameters in the action. Eight of these correspond to the eight Thurston geometries, while the ninth describes the metric of a squashed three sphere. It therefore appears that one can c…

Nonholonomic Clifford and Finsler Structures, Non-Commutative Ricci Flows, and Mathematical Relativity

In this summary of Habilitation Thesis, it is outlined author's 18 years research activity on mathematical physics, geometric methods in particle physics and gravity, modifications and applications (after defending his PhD thesis in 1994). Ten most relevant publications are structured conventionally into three "strategic directions": 1) nonholonomic geometric flows evolutions and exact solutions for Ricci solitons and field equations in (modified) gravity theories; 2) geometric methods in quantization of models with nonlinear dynamics and anisotropic field interactions; 3) (non) commutative ge…

关于推广Ricci流的一类抛物型曲率下界

On a parabolic curvature lower bound generalizing Ricci flows

针对超Ricci流无法保留Perelman泛函单调性的问题，本文用最优传输刻画了Buzano提出的推广条件，证明其等价于多种分析不等式，且多数证明无需张量分析，为奇异情形推广提供可能。

The Ricci flow of asymptotically hyperbolic mass and applications

We consider the evolution of the asymptotically hyperbolic mass under the curvature-normalized Ricci flow of asymptotically hyperbolic, conformally compactifiable manifolds. In contrast to asymptotically flat manifolds, for which ADM mass is constant during Ricci flow, we show that the mass of an asymptotically hyperbolic manifold of dimension n>2 decays smoothly to zero exponentially in the flow time. From this, we obtain a no-breathers theorem and a Ricci flow based, modified proof of the scalar curvature rigidity of zero-mass asymptotically hyperbolic manifolds. We argue that the nonconstan…

扩展 Ricci 流背景中的平均曲率流

Mean curvature flow in an extended Ricci flow background

本文研究扩展 Ricci 流背景下的平均曲率流，分析加权 GHY 作用量及其变分与演化，推导几何量方程，并建立平均曲率孤立子的单调性公式、构造与刻画。

Complex optimal transport and the pluripotential theory of Kähler-Ricci solitons

Let (X,L) be a (semi-) polarized complex projective variety and T a real torus acting holomorphically on X with moment polytope P. Given a probability density g on P we introduce a new type of Monge-Ampere measure on X, defined for singular T-invariant metrics on the line bundle L, generalizing the ordinary Monge-Ampere of global pluripotential theory, which corresponds to the case when T is trivial (or g=1). In the opposite extreme case when T has maximal rank, i.e. (X,L,T) is a toric variety, the solution of the corresponding Monge-Ampere equation with right hand side μcorresponds to the con…