Hypersurfaces of constant curvature in hyperbolic space

Fei-tsen Liang

Fei-tsen Liang Associate Researcher, Institiute of Mathematics, Acdemia Sinica, Taipei, Taiwan

Keywords: constant Gauss curvature, prescribed boundary.

Abstract

We continue the work done in [2],[3] which investigates the problem of finding Weingarten hypersurfaces of constant curvature satisfying (1), (2) below in hyperbolic space $ H^{n+1}$ with a prescribed asymptotic boundary at infinity.
In [2], the focus is on the case of finding complete hypersurfaces with positive hyperbolic principal curvatures everywhere; in [3], the focus is on finding graphs over a domain with nonnegative mean curvature.
In [2] and [3], some restriction is imposed on $\sigma$ to assure us of the existence. The main aim of this article is to remove these restrictions.
The results stated in the manuscript, as well as more general ones
have been proved in [4] and [5] with a less elementary approach.

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