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arXiv · 2609.01104

The asymptotic Plateau problem for Hypersurfaces of constant $H_{k}$ curvature in hyperbolic space

Xinqun Mei·Jin Yan
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Abstract

In this paper, we study the asymptotic Plateau problem in hyperbolic space for hypersurfaces of constant $H_k$-curvature. We prove the existence of a smooth complete $k$-convex hypersurface in $\mathbb{H}^{n+1}$ satisfying \[ H_k(\kappa)=\sigma, \qquad \sigma\in(0,1), \] with prescribed asymptotic boundary at infinity. In particular, our result extends the range of the constant $\sigma$ in the existence theorem of Guan and Spruck [J. Eur. Math. Soc. 12 (2010), no. 3, 797--817] for $H_{k}$ curvature to the full interval $(0,1)$.

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Xinqun Mei, Jin Yan. 2026-09-01. The asymptotic Plateau problem for Hypersurfaces of constant $H_{k}$ curvature in hyperbolic space. https://arxiv.org/abs/2609.01104

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