Asymptotic Plateau problem for prescribed mean curvature hypersurfaces

Casteras, Jean-Baptiste; Holopainen, Ilkka; Ripoll, Jaime

doi:10.1090/proc/14829

Asymptotic Plateau problem for prescribed mean curvature hypersurfaces
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by Jean-Baptiste Casteras, Ilkka Holopainen and Jaime B. Ripoll PDF

Proc. Amer. Math. Soc. 148 (2020), 1731-1743 Request permission

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.

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