Tokyo Journal of Mathematics

Maximum Principle and Convergence of Fundamental Solutions for the Ricci Flow

Shu-Yu HSU

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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 Neumann Green function of the conjugate heat equation for the Ricci flow in $B_k\times (0,T)$ to the minimal fundamental solution of the conjugate heat equation as $k\to\infty$. We will prove the uniqueness of the fundamental solution under some exponential decay assumption on the fundamental solution. We will also give a detail proof of the convergence of the fundamental solutions of the conjugate heat equation for a sequence of pointed Ricci flow $(M_k\times (-\alpha,0],x_k,g_k)$ to the fundamental solution of the limit manifold as $k\to\infty$ which was used without proof by Perelman in his proof of the pseudolocality theorem for Ricci flow [P].

Article information

Tokyo J. Math., Volume 32, Number 2 (2009), 501-516.

First available in Project Euclid: 22 January 2010

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Zentralblatt MATH identifier

Primary: 58J35: Heat and other parabolic equation methods
Secondary: 53C43: Differential geometric aspects of harmonic maps [See also 58E20]


HSU, Shu-Yu. Maximum Principle and Convergence of Fundamental Solutions for the Ricci Flow. Tokyo J. Math. 32 (2009), no. 2, 501--516. doi:10.3836/tjm/1264170246.

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