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

作者：Song, Xingyu; Wu, Ling; Zhu, Meng^*

来源：Journal of Geometry and Physics, 2023, 194: 104997.

DOI：10.1016/j.geomphys.2023.104997

• Bakry-emery Ricci
• Laplace-Beltrami operator
• Heat kernel
• Liouville theorem
• Volume comparison
• Eigenvalue estimate

We establish a Gaussian upper bound of the heat kernel for the Laplace-Beltrami operator on complete Riemannian manifolds with Bakry-emery 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-emery 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.