A splitting theorem for capillary graphs under Ricci lower bounds
| Autor: | Luciano Mari, Giulio Colombo, Marco Rigoli |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Mathematics - Differential Geometry
Pure mathematics Mean curvature 010102 general mathematics Riemannian manifold Lipschitz continuity 01 natural sciences Dirichlet distribution Primary 53C21 53C42 Secondary 53C24 58J65 31C12 31B05 symbols.namesake Mathematics - Analysis of PDEs Corollary Differential Geometry (math.DG) 0103 physical sciences symbols FOS: Mathematics Splitting theorem Product topology 010307 mathematical physics Mathematics::Differential Geometry 0101 mathematics Analysis Ricci curvature Mathematics Analysis of PDEs (math.AP) |
| DOI: | 10.48550/arxiv.2007.15143 |
| Popis: | In this paper, we study capillary graphs defined on a domain $\Omega$ of a complete Riemannian manifold $M$, where a graph is said to be capillary if it has constant mean curvature and locally constant Dirichlet and Neumann conditions on $\partial \Omega$. Our main result is a splitting theorem both for $\Omega$ and for the graph function on a class of manifolds with nonnegative Ricci curvature. As a corollary, we classify capillary graphs over domains that are globally Lipschitz epigraphs or slabs in a product space $M = N \times \mathbb{R}$, where $N$ has slow volume growth and non-negative Ricci curvature, including the case $M = \mathbb{R}^2,\mathbb{R}^3$. A technical core of the paper is a new gradient estimate for positive CMC graphs on manifolds with Ricci lower bounds. Comment: 42 pages. Bibliography updated. Accepted on J. Funct. Anal |
| Databáze: | OpenAIRE |
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