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arXiv · 1605.01935

Dirichlet problem for $f$-minimal graphs

Jean-Baptiste Casteras·Esko Heinonen·Ilkka Holopainen
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Abstract

We study the asymptotic Dirichlet problem for $f$-minimal graphs in Cartan-Hadamard manifolds $M$. $f$-minimal hypersurfaces are natural generalizations of self-shrinkers which play a crucial role in the study of mean curvature flow. In the first part of this paper, we prove the existence of $f$-minimal graphs with prescribed boundary behavior on a bounded domain $Ω\subset M$ under suitable assumptions on $f$ and the boundary of $Ω$. In the second part, we consider the asymptotic Dirichlet problem. Provided that $f$ decays fast enough, we construct solutions to the problem. Our assumption on the decay of $f$ is linked with the sectional curvatures of $M$. In view of a result of Pigola, Rigoli and Setti, our results are almost sharp.

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Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen. 2019-07-25. Dirichlet problem for $f$-minimal graphs. https://doi.org/10.1007/s11854-019-0051-5

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