相关论文: Notes on Perelman's papers

We discuss some of the key ideas of Perelman's proof of Poincar\'e's conjecture via the Hamilton program of using the Ricci flow, from the perspective of the modern theory of nonlinear partial differential equations.

In this paper we present several curvature estimates and convergence results for solutions of the Ricci flow. The curvature estimates depend on smallness of certain local space-time integrals of the norm of the Riemann curvature tensor,…

This paper introduces a functional generalizing Perelman's weighted Hilbert-Einstein action and the Dirichlet energy for spinors. It is well-defined on a wide class of non-compact manifolds; on asymptotically Euclidean manifolds, the…

We announce a new proof of the uniform estimate on the curvature of solutions to the Ricci flow on a compact K\"ahler manifold $M^n$ with positive bisectional curvature. In contrast to the recent work of X. Chen and G. Tian, our proof of…

It is the purpose of this article to establish a technical tool to study regularity of solutions to parabolic equations on manifolds. As applications of this technique, we prove that solutions to the Ricci-DeTurck flow, the surface…

We analyze the Ricci flow of a noncompact metric that describes a two-dimensional black hole. We consider entanglement entropy of a 2d black hole which is due to the quantum correlations between two subsystems: one is inside and the other…

For an immortal Ricci flow on an $m$-dimensional $(m\ge 3)$ closed manifold, we show the following convergence results: (1) if the curvature and diameter are uniformly bounded, then any unbounded sequence of time slices sub-converges to a…

In this note, we give a new proof for Perelman's scalar curvature and diameter estimates for the K\"ahler-Ricci flow on Fano manifolds. The proof relies on a new Harnack estimate for a special family of functions in space-time. Our new…

We prove sharp lower bounds for eigenvalues of the drift Laplacian for a modified Ricci flow. The modified Ricci flow is a system of coupled equations for a metric and weighted volume that plays an important role in Ricci flow. We will also…

We investigate a new geometric flow which consists of a coupled system of the Ricci flow on a closed manifold M with the harmonic map flow of a map phi from M to some closed target manifold N with a (possibly time-dependent) positive…

In this work, we study and solve the normalized Ricci flow equation for circle bundles over surfaces. Moreover, we study the asymptotic behavior of the solutions and their connections to some model geometries.

This work consists an introduction to the classical and quantum information theory of geometric flows of (relativistic) Lagrange--Hamilton mechanical systems. Basic geometric and physical properties of the canonical nonholonomic…

In this short notes, we discuss monotonicity formulas under various rescaled versions of Ricci flow. The main result is Theorem \ref{theo rescaled}.

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

We consider a closed manifold M with a Riemannian metric g(t) evolving in direction -2S(t) where S(t) is a symmetric two-tensor on (M,g(t)). We prove that if S satisfies a certain tensor inequality, then one can construct a forwards and a…

In this paper we study the Ricci flow on surfaces homeomorphic to a cylinder (that is, a product of the circle with a compact interval). We prove longtime existence results, results on the asymptotic behavior of the flow, and we report on…

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…

We report on some recent progress achieved in [arXiv:2111.14811] on the ergodicity of the frame flow of negatively-curved Riemannian manifolds. We explain the new ideas leading to ergodicity for nearly $0.25$-pinched manifolds and give…

We study the behavior of the normalized Ricci flow of invariant Riemannian homogeneous metrics at infinity for generalized Wallach spaces, generalized flag manifolds with four isotropy summands and second Betti number equal to one, and the…

In this paper, we prove Perelman type $\mathcal{W}$-entropy formulae and global differential Harnack estimates for positive solutions to porous medium equation on the closed Riemannian manifolds with Ricci curvature bounded below. As…