# Asymptotic Plateau problem for prescribed mean curvature hypersurfaces

## TL;DR

This paper establishes the existence of hypersurfaces with prescribed mean curvature in certain negatively curved manifolds, solving an asymptotic boundary value problem with conditions on curvature and boundary data.

## Contribution

It proves the existence of solutions to the asymptotic Plateau problem for prescribed mean curvature hypersurfaces in Cartan-Hadamard manifolds under specific curvature and boundary conditions.

## Key findings

- Existence of solutions under strict convexity and curvature bounds
- Construction of hypersurfaces with prescribed mean curvature
- Multiplicity results in low-dimensional cases

## Abstract

We prove the existence of solutions to the asymptotic Plateau problem for hypersurfaces of prescribed mean curvature in Cartan-Hadamard manifolds $N$. More precisely, given a suitable subset $L$ of the asymptotic boundary of $N$ and a suitable function $H$ on $N$, we are able to construct a set of locally finite perimeter whose boundary has generalized mean curvature $H$ provided that $N$ satisfies the so-called strict convexity condition and that its sectional curvatures are bounded from above by a negative constant. We also obtain a multiplicity result in low dimensions.