Abstract
In this note, we obtain the rigidity of the sharp Cheng-Yau gradient estimate for positive harmonic functions on surfaces with non-negative 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 non-negative Ricci curvature. Moreover, we also obtain the corresponding stability results.
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Acknowledgements
Q. Hu and G. Xu thank the Department of Mathematics, Shantou University for hosting their visit to Shantou, part of the work was done during the visit. Research was partially supported by Beijing Natural Science Foundation Z190003, NSFC 11771230 and NSFC 12141103. Research was partially supported by GDNSF 2021A1515010264 and NSFC 11571215.
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Dedicated to Professor Peter Li on the occasion of his 70th birthday.
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Hu, Q., Xu, G. & Yu, C. The Rigidity and Stability of Gradient Estimates. J Geom Anal 32, 279 (2022). https://doi.org/10.1007/s12220-022-01022-x
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DOI: https://doi.org/10.1007/s12220-022-01022-x