Calculus of Variations and Geometric Measure Theory

G. Colombo - L. Mari - M. Rigoli

A splitting theorem for capillary graphs under Ricci lower bounds

created by mari1 on 25 Nov 2020
modified on 30 Aug 2021



Inserted: 25 nov 2020
Last Updated: 30 aug 2021

Journal: J. Funct. Anal.
Volume: 281
Number: 8
Pages: 50
Year: 2021

ArXiv: 2007.15143 PDF


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 all 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. A technical core of the paper is a new gradient estimate for positive CMC graphs on manifolds with Ricci lower bounds, of independent interest.