A splitting theorem for capillary graphs under Ricci lower bounds

Abstract

In this paper, we study capillary graphs defined on a domain Ω of a complete Riemannian manifold, where a graph is said to be capillary if it has constant mean curvature and locally constant Dirichlet and Neumann conditions on ∂Ω. Our main result is a splitting theorem both for Ω 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 N×R, where N has slow volume growth and non-negative Ricci curvature. A technical core of the paper is a new gradient estimate for positive CMC graphs on manifolds with Ricci lower bounds.