Heat kernel estimate for the Laplace-Beltrami operator under Bakry-Émery Ricci curvature condition and applications

2023/06/23 by Song, Xingyu, Wu, Ling, Zhu, Meng · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2306.13261

Abstract

We establish a Gaussian upper bound of the heat kernel for the Laplace-Beltrami operator on complete Riemannian manifolds with Bakry-Émery Ricci curvature bounded below. As applications, we first prove an L¹-Liouville property for non-negative subharmonic functions when the potential function of the Bakry-Émery Ricci curvature tensor is of at most quadratic growth. Then we derive lower bounds of the eigenvalues of the Laplace-Beltrami operator on closed manifolds. An upper bound of the bottom spectrum is also obtained.

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• Gradient estimates for (p,V)-harmonic functions on Riemannian manifolds

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