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arXiv · 2401.14097

Prescribed mean curvature hypersurfaces in conformal product manifolds

Qiang Gao·Hengyu Zhou
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Abstract

In this paper, we establish the existence of prescribed mean curvature (PMC) hypersurfaces in conformal product manifolds with (possibly empty) $C^{1,\alpha}$ fixed graphical boundaries under a barrier condition. This result generalizes Gerhardt's work to non-flat conformal backgrounds. As a consequence, we obtain new solutions to the high-dimensional PMC Plateau problem with explicitly characterized topology. Moreover, under a quasi-decreasing condition on the PMC function, we demonstrate that the resulting hypersurfaces are $C^1$ graphs.

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Qiang Gao, Hengyu Zhou. 2024-01-25. Prescribed mean curvature hypersurfaces in conformal product manifolds. https://arxiv.org/abs/2401.14097

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