arXiv research

The paper studies mean curvature flow in a Ricci flow background with extended Ricci flow.

The paper studies geometric constants under modified Ricci flows with variable parameters.

We simplify and improve the curvature estimates in the paper: On the conditions to extend Ricci flow(II). Furthermore, we develop some volume estimates for the Ricci flow with bounded scalar curvature. These estimates can be applied to study the singularities of the Ricci flow and convergence properties of the Kähler R…

We consider four extended Ricci flow systems---that is, Ricci flow coupled with other geometric flows---and prove dynamical stability of certain classes of stationary solutions of these flows. The systems include Ricci flow coupled with harmonic map flow (studied abstractly and in the context of Ricci flow on warped pr…

Extends heat kernel estimates for super Ricci flow.

Study applies Huisken formula to mean curvature flow in Ricci soliton background.

Combinatorial Ricci flow finds hyperbolic metrics on 3-manifolds.

Along a Ricci flow solution on a closed manifold, we show that if Ricci curvature is uniformly bounded from below, then a scalar curvature integral bound is enough to extend flow. Moreover, this integral bound condition is optimal in some sense.

Theory of Ricci flow for Courant algebroids, preserving isometry group.

Smooth 3D flows from non-smooth starting points.

The paper applies combinatorial Ricci flows to prove hyperbolic structures on 3-manifolds.

The paper extends entropy formulas to super Ricci flows on metric measure spaces.

In this paper, we study the singularities of two extended Ricci flow systems --- connection Ricci flow and Ricci harmonic flow using newly-defined curvature quantities. Specifically, we give the definition of three types of singularities and their corresponding singularity models, and then prove the convergence. In add…

Extends Ricci flow theory to Kato-type curvature bounds, proving manifold properties.

Study extends convergence results to noncompact Ricci flows.

In this paper, we derive a Sobolev inequality along an extended Ricci flow and prove a point-wise Guassian type bound for the fundamental solutions of the conjugate heat equation under the flow.

Global regularity proved for 4D Ricci flow with scalar curvature integral bound.

Proves uniqueness of Ricci flow with scaling invariant estimates.

We characterize the conjugate linearized Ricci flow and the associated backward heat kernel on closed three--manifolds of bounded geometry. We discuss their properties, and introduce the notion of Ricci flow conjugated constraint sets which characterizes a way of Ricci flow averaging metric dependent geometrical data. …

Study shows uniform bounds on torsion and curvature for Chern-Ricci flow solutions.

We discuss from a geometric point of view the connection between the renormalization group flow for non--linear sigma models and the Ricci flow. This offers new perspectives in providing a geometrical landscape for 2D quantum field theories. In particular we argue that the structure of Ricci flow singularities suggests…

The paper studies Ricci flow with finite curvature integrals on manifolds.

Paper shows regularizing flow for conical Kähler-Ricci equations.

In this paper we study the long time existence of the Ricci-harmonic flow in terms of scalar curvature and Weyl tensor which extends Cao's result \cite{Cao2011} in the Ricci flow. In dimension four, we also study the integral bound of the "Riemann curvature" for the Ricci-harmonic flow generalizing a recently result of…

Study explores how scalar functionals evolve under Ricci flow.

We prove several differential Harnack inequalities for positive solutions to nonlinear backward heat equations with different potentials coupled with the Ricci flow. We also derive an interpolated Harnack inequality for the nonlinear heat equation under the $\varepsilon$-Ricci flow on a closed surface. These new Harnac…

The paper extends Ricci flow theory with Type-I scalar curvature bounds, proving entropy convergence and characterizing singular sets.

Let $Δ_\varphi = Δ-\nabla \varphi \nabla$ be a symmetric diffusion operator with an invariant weighted volume measure $dμ= e^{-\varphi} dv$ on an $n$-dimensional compact Riemannian manifold $(M,g)$, where $g=g(t)$ solves the extended Ricci flow. In this article we study the evolution and monotonicty of the first nonzer…

We present two new conditions to extend the Ricci flow on a compact manifold over a finite time, which are improvements of some known extension theorems.

We propose the time-dependent generalization of an `ordinary' autonomous human biomechanics, in which total mechanical + biochemical energy is not conserved. We introduce a general framework for time-dependent biomechanics in terms of jet manifolds derived from the extended musculo-skeletal configuration manifold. The …

In this survey article, we discuss some topics on self-similar solutions to the Ricci flow and the mean curvature flow. Self-similar solutions to the Ricci flow are known as Ricci solitons. In the first part of this paper we discuss a lower diameter bound for compact manifolds with shrinking Ricci solitons. Such a boun…

Assume $M$ is a closed 3-manifold whose universal covering is not $S^3$. We show that the obstruction to extend the Ricci flow is the boundedness $L^{3/2}$-norm of the scalar curvature $R(t)$, i.e, the Ricci flow can be extended over time $T$ if and only if the $||R(t)||_{L^{3/2}}$ is uniformly bounded for $0 \leq t < …

In this paper, we investigate the Gauss maps of a Ricci-mean curvature flow. A Ricci-mean curvature flow is a coupled equation of a mean curvature flow and a Ricci flow on the ambient manifold. Ruh and Vilms proved that the Gauss map of a minimal submanifold in a Euclidean space is a harmonic map, and Wang extended thi…

Study higher-dimensional Ricci flow solutions, proving uniqueness.

In this short note we announce a regularity theorem for Kähler-Ricci flow on a compact Fano manifold (Kähler manifold with positive first Chern class) and its application to the limiting behavior of Kähler-Ricci flow on Fano 3-manifolds. Moreover, we also present a partial $C^0$ estimate to the Kähler-Ricci flow under …

Paper proves Luo's conjecture for 3D triangulated manifolds.

Ricci flow can change metrics with intermediate curvatures.

The paper extends Perelman's theorems on Ricci flow entropy.

In this note we obtain local derivative estimates of Shi-type for the heat equation coupled to the Ricci flow. As applications, in part combining with Kuang's work, we extend some results of Zhang and Bamler-Zhang including distance distortion estimates and a backward pseudolocality theorem for Ricci flow on compact ma…

In this paper, we study monotonicity formulas of eigenvalues and entropies along the rescaled List's extended Ricci flow. We derive some monotonicity formulas of eigenvalues of Laplacian which generalize those of Li in [8] and Cao-Hou-Ling in [3]. Moreover, we also consider monotonicity formulas of $\mathcal{F}_k$-func…

Derives $L^{\infty}$ estimate for Kähler-Ricci flows with weaker conditions.

The paper studies deformation of discrete conformal structures on surfaces using combinatorial curvature flows.

The paper introduces combinatorial curvature and flow for polyhedral surfaces, proving rigidity and solving the Yamabe problem.

Extends Margulis Lemma to RCD(K,N) spaces.

Paper improves heat kernel estimates on Ricci shrinkers.

The Kähler-Ricci flow's singularities are analyzed with bounds and convergence results.

Ricci flows with bounded scalar curvature cannot develop Type I singular points.

Consider the unnormalized Ricci flow $(g_{ij})_t = -2R_{ij}$ for $t\in [0,T)$, where $T < \infty$. Richard Hamilton showed that if the curvature operator is uniformly bounded under the flow for all times $t\in [0,T)$ then the solution can be extended beyond $T$. We prove that if the Ricci curvature is uniformly bounded…