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Neumann Li-Yau gradient estimate under integral Ricci curvature bounds


Author: Xavier Ramos Olivé
Journal: Proc. Amer. Math. Soc. 147 (2019), 411-426
MSC (2010): Primary 58J32, 58J35
DOI: https://doi.org/10.1090/proc/14213
Published electronically: September 17, 2018
MathSciNet review: 3876759
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Abstract: We prove a Li-Yau gradient estimate for positive solutions to the heat equation, with Neumann boundary conditions, on a compact Riemannian submanifold with boundary $ {\bf M}^n\subseteq {\bf N}^n$, satisfying the integral Ricci curvature assumption:

$\displaystyle D^2 \sup _{x\in {\bf N}} \left ( \oint _{B(x,D)} \vert Ric^-\vert^p dy \right )^{\frac {1}{p}} < K$ (1)

for $ K(n,p)$ small enough, $ p>n/2$, and $ diam({\bf M})\leq D$. The boundary of $ {\bf M}$ is not necessarily convex, but it needs to satisfy the interior rolling $ R-$ball condition.

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Additional Information

Xavier Ramos Olivé
Affiliation: Department of Mathematics, University of California, Riverside, Riverside, California 92521
Email: olive@math.ucr.edu

DOI: https://doi.org/10.1090/proc/14213
Keywords: Geometric analysis, differential geometry, Neumann heat kernel, integral Ricci curvature, Li-Yau gradient estimate
Received by editor(s): April 12, 2018
Received by editor(s) in revised form: April 25, 2018
Published electronically: September 17, 2018
Communicated by: Guofang Wei
Article copyright: © Copyright 2018 American Mathematical Society