*Published Paper*

**Inserted:** 25 nov 2020

**Last Updated:** 17 aug 2024

**Journal:** J. Funct. Anal.

**Volume:** 281

**Number:** 8, Paper No. 109136

**Pages:** 50

**Year:** 2021

**Doi:** 10.1016/j.jfa.2021.109136

**Abstract:**

In this paper, we study capillary graphs defined on a domain $\Omega$ 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 $\partial \Omega$. 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 $N \times \mathbb{R}$, where $N$ has slow volume growth and non-negative Ricci curvature, including the cases $N\times \mathbb{R} = \mathbb{R}^2$ and $\mathbb{R}^3$. A technical core of the paper is a new gradient estimate for positive CMC graphs on manifolds with Ricci lower bounds, of independent interest.