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Dirichlet problem for \(f\)-minimal graphs. (English) Zbl 1427.53079

Summary: 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 \(\Omega \subset M\) under suitable assumptions on \(f\) and the boundary of \(\Omega\). 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 S. Pigola et al. [Pac. J. Math. 206, No. 1, 195–217 (2002; Zbl 1057.53049)], our results are almost sharp.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Citations:

Zbl 1057.53049
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References:

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