Convergence of fundamental solutions of linear parabolic equations under Cheeger–Gromov convergence

  title={Convergence of fundamental solutions of linear parabolic equations under Cheeger–Gromov convergence},
  author={Peng Lu},
  journal={Mathematische Annalen},
  • P. Lu
  • Published 6 May 2010
  • Mathematics
  • Mathematische Annalen
In this note we show the convergence of the fundamental solutions of the parabolic equations assuming the Cheeger–Gromov convergence of the underlying manifolds and the uniform L1-bound of the solutions. We also prove a local integral estimate of fundamental solutions. 
Ancient Ricci flows with asymptotic solitons
We study ancient Ricci flows which admit asymptotic solitons in the sense of Perelman [Per02, Proposition 11.2]. We prove that the asymptotic solitons must coincide with Bamler’s tangent flows at
On the equivalence between noncollapsing and bounded entropy for ancient solutions to the Ricci flow
Abstract As a continuation of a previous paper, we prove Perelman’s assertion, that is, for ancient solutions to the Ricci flow with bounded nonnegative curvature operator, uniformly bounded entropy
Brownian motion on stationary random manifolds
We introduce the concept of a stationary random manifold with the objective of treating in a unified way results about manifolds with transitive isometry group, manifolds with a compact quotient, and
Entropy, noncollapsing, and a gap theorem for ancient solutions to the Ricci flow
In this paper we discuss the asymptotic entropy for ancient solutions to the Ricci flow. We prove a gap theorem for ancient solutions, which could be regarded as an entropy counterpart of Yokota's


The convergence of the minimal positive fundamental solutions under Ricci flow
In an unpublished paper, Hsu gives a proof of the convergence of the fundamental solutions. Since we had a problem understanding Hsu's paper, in this paper we give a detailed proof of the convergence
Maximum principles introduction to the theory of weak solutions Holder estimates existence, uniqueness and regularity of solutions further theory of weak solutions strong solutions fixed point
Maximum Principle and Convergence of Fundamental Solutions for the Ricci Flow
In this paper we will prove a maximum principle for the solutions of linear parabolic equation on complete non-compact manifolds with a time varying metric. We will prove the convergence of the
On the parabolic kernel of the Schrödinger operator
Etude des equations paraboliques du type (Δ−q/x,t)−∂/∂t)u(x,t)=0 sur une variete riemannienne generale. Introduction. Estimations de gradients. Inegalites de Harnack. Majorations et minorations des
Hamilton's gradient estimate for the heat kernel on complete manifolds
In this paper we extend a gradient estimate of R. Hamilton for positive solutions to the heat equation on closed manifolds to bounded positive solutions on complete, non-compact manifolds with Re ≥
Pseudolocality for the Ricci Flow and Applications
Abstract Perelman established a differential Li-Yau-Hamilton $\left( \text{LHY} \right)$ type inequality for fundamental solutions of the conjugate heat equation corresponding to the Ricci flow on
Some gradient estimates for the heat equation on domains and for an equation by Perelman
In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local Li-Yau gradient estimate that matches the global
The entropy formula for the Ricci flow and its geometric applications
We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric
Notes on Perelman's papers
These are detailed notes on Perelman's papers "The entropy formula for the Ricci flow and its geometric applications" and "Ricci flow with surgery on three-manifolds".
The Ricci Flow: Techniques and Applications: Part II: Analytic Aspects
Contents Preface ix What Part II is about ix Highlights and interdependences of Part II xi Acknowledgments xiii Contents of Part II of Volume Two xvii Notation and Symbols xxiii Chapter 10. Weak