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arXiv · 1801.04210

Asymptotic Dirichlet problems in warped products

Jean-Baptiste Casteras·Esko Heinonen·Ilkka Holopainen·Jorge H. de Lira
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Abstract

We study the asymptotic Dirichlet problem for Killing graphs with prescribed mean curvature $H$ in warped product manifolds $M\times_\varrho \mathbb{R}$. In the first part of the paper, we prove the existence of Killing graphs with prescribed boundary on geodesic balls under suitable assumptions on $H$ and the mean curvature of the Killing cylinders over geodesic spheres. In the process we obtain a uniform interior gradient estimate improving previous results by Dajczer and de Lira. In the second part we solve the asymptotic Dirichlet problem in a large class of manifolds whose sectional curvatures are allowed to go to $0$ or to $-\infty$ provided that $H$satisfies certain bounds with respect to the sectional curvatures of $M$ and the norm of the Killing vector field. Finally we obtain non-existence results if the prescribed mean curvature function $H$ grows too fast.

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Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen, Jorge H. de Lira. 2019-05-29. Asymptotic Dirichlet problems in warped products. https://doi.org/10.1007/s00209-019-02346-1

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