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Geometria Diferencial

Asymptotic Plateau problem for prescribed mean curvature hypersurfaces
Ilkka Holopainen

University of Helsinki
Terça-feira, 26 de novembro de 2019, 15:30
Sala 236

I will talk on a recent joint paper with Jean-Baptiste Casteras and Jaime Ripoll.

Let $N$ be an $n$-dimensional Cartan--Hadamard manifold that satisfies the so-called strict convexity condition and has strictly negative upper bound for sectional curvatures, $K\le-\alpha^2<0$. Given a suitable subset $L\subset\partial_\infty N$ of the asymptotic boundary of $N$ and a continuous function $H\colon N\to [-H_0,H_0],\ H_0<(n-1)\alpha$, we prove the existence of an open subset $Q\subset N$ of locally finite perimeter whose boundary $M$ has generalized mean curvature $H$ towards $N\setminus Q$ and $\partial_\infty M=L$. By regularity theory, $M$ is a $C^2$-smooth $(n-1)$-dimensional submanifold up to a closed singular set of Hausdorff dimension at most $n-8$. In particular, $M$ is $C^2$-smooth if $n\le 7$. Moreover, if $H\in [-H_0,H_0]$ is constant and $n\le 7$, there are at least two disjoint hypersurfaces $M_1$, $M_2$ with constant mean curvature $H$ and $\partial_\infty M_i=L,\ i=1,2$.

Our results generalize those of Alencar and Rosenberg, Tonegawa, and others.