A splitting theorem for capillary graphs under Ricci lower bounds

Abstract

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.

arXiv comment42 pages. Bibliography updated. Accepted on J. Funct. Anal

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Cited by in corpus (1)

• Recent rigidity results for graphs with prescribed mean curvature