Jenkins-Serrin graphs in $M\times\mathbb{R}$

TL;DR

This paper surveys the development of Jenkins-Serrin problems for prescribed mean curvature graphs on Riemannian manifolds, highlighting existence results, geometric restrictions, and new theorems including translating solutions.

Contribution

It provides a comprehensive survey of Jenkins-Serrin problems on Riemannian manifolds and introduces new existence theorems for translating Jenkins-Serrin graphs.

Findings

Established geometric restrictions on boundary domains.

Proved existence of translating Jenkins-Serrin graphs.

Improved previous results with new theorems.

Abstract

The so called Jenkins-Serrin problem is a kind of Dirichlet problem for graphs with prescribed mean curvature that combines, at the same time, continuous boundary data with regions of the boundary where the boundary values explodes either to $+\infty$ or to $-\infty.$ We give a survey on the development of Jenkins-Serrin type problems over domains on Riemannian manifolds. The existence of this type of graphs imposes restrictions on the geometry of the boundary of these domains. We also improve some earlier results by proving Theorem 1.8, and prove the existence of translating Jenkins-Serrin graphs (Theorem 1.9).