A note on local gradient estimate on Alexandrov spaces
Bobo Hua, Chao Xia
TL;DR
This paper establishes a refined local gradient estimate for harmonic functions on Alexandrov spaces with Ricci curvature bounds, improving previous results especially for negative Ricci bounds using advanced iteration and Bochner formulas.
Contribution
It introduces a refined Moser iteration method combined with Zhang-Zhu's Bochner formula to improve gradient estimates on Alexandrov spaces with Ricci curvature bounds.
Findings
Proves Cheng-Yau type gradient estimate for harmonic functions on Alexandrov spaces.
Enhances previous estimates for spaces with negative Ricci lower bounds.
Utilizes a refined Moser iteration and Bochner formula for the proof.
Abstract
In this note, we prove Cheng-Yau type local gradient estimate for harmonic functions on Alexandrov spaces with Ricci curvature bounded below. We adopt a refined version of Moser's iteration which is based on Zhang-Zhu's Bochner type formula. Our result improves the previous one of Zhang-Zhu in the case of negative Ricci lower bound.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Nonlinear Partial Differential Equations · Numerical methods in inverse problems