Asymptotic Plateau problem for prescribed mean curvature hypersurfaces

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.

The proof of Lemma 2.1 is rewritten providing a quantitative estimate for the decay of the functional $\mathcal{F}$. To appear in Proceedings of the AMS

References in corpus (2)

• Convexity at infinity in Cartan-Hadamard manifolds and applications to the asymptotic Dirichlet and Plateau problems
• Embedded $H$-Planes in Hyperbolic 3-Space