Asymptotic Dirichlet problems in warped products

TL;DR

This paper investigates the existence and non-existence of Killing graphs with prescribed mean curvature in warped product manifolds, providing new gradient estimates and solving the asymptotic Dirichlet problem under various curvature conditions.

Contribution

It introduces new existence results for Killing graphs with prescribed boundary and mean curvature, along with improved gradient estimates and non-existence results based on curvature and growth conditions.

Findings

Existence of Killing graphs with prescribed boundary under certain curvature assumptions.

Improved uniform interior gradient estimates for solutions.

Non-existence results when mean curvature grows too fast.

Abstract

We study the asymptotic Dirichlet problem for Killing graphs with prescribed mean curvature $H$ in warped product manifolds $M\times_\varrho \mathbb{R}$. In the first part of the paper, we prove the existence of Killing graphs with prescribed boundary on geodesic balls under suitable assumptions on $H$ and the mean curvature of the Killing cylinders over geodesic spheres. In the process we obtain a uniform interior gradient estimate improving previous results by Dajczer and de Lira. In the second part we solve the asymptotic Dirichlet problem in a large class of manifolds whose sectional curvatures are allowed to go to $0$ or to $-\infty$ provided that $H$satisfies certain bounds with respect to the sectional curvatures of $M$ and the norm of the Killing vector field. Finally we obtain non-existence results if the prescribed mean curvature function $H$ grows too fast.