The rigidity and stability of gradient estimates
Qixuan Hu, Guoyi Xu, Chengjie Yu
TL;DR
This paper establishes the rigidity and stability of key gradient estimates for harmonic functions and heat equations on surfaces and manifolds with nonnegative curvature, enhancing understanding of geometric analysis.
Contribution
It proves the rigidity and stability of sharp gradient estimates for harmonic functions and heat equations on manifolds with nonnegative curvature, extending previous results.
Findings
Rigidity of the Cheng-Yau gradient estimate for harmonic functions.
Rigidity of the Li-Yau gradient estimate for heat equations.
Stability results for related Green's function estimates.
Abstract
In this note, we obtain the rigidity of the sharp Cheng-Yau gradient estimate for positive harmonic functions on surfaces with nonegative Gaussian curvature, the rigidity of the sharp Li-Yau gradient estimate for positive solutions to heat equations and the related estimates for Dirichlet Green's functions on Riemannian manifolds with nonnegative Ricci curvature. Moreover, we also obtain the corresponding stability results.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Nonlinear Partial Differential Equations · Geometry and complex manifolds