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On the existence of convex hypersurfaces of constant Gauss curvature in hyperbolic space. (English) Zbl 0823.53047
Let $$M$$ be a complete embedded $$K$$-hypersurface of a hyperbolic $$(n+1)$$- space $$H^{n+1}$$, that is the Gauss-Kronecker curvature of $$M$$ is the constant $$K = K_{\text{ext.}} - 1$$ (sic), where $$K_{\text{ext.}}$$ is the determinant of the second fundamental form. The authors prove that a codimension-one embedded submanifold $$\Gamma$$ of $$\partial_ \infty (H^{n + 1})$$ is the asymptotic boundary of such an $$M$$ for any $$K \in (-1,0)$$. The authors’ approach is to construct the desired $$M$$ as the limit of $$K$$-graphs over a fixed compact domain in a horosphere for appropriate boundary data. Thus an important part of their study is an existence theory for $$K$$-hypersurfaces which are graphs over a bounded domain in a horosphere. This is accomplished by solving a Monge-Ampère equation for the Gauss curvature using the recent work of B. Guan and the second author [Ann. Math., II. Ser. 138, No. 3, 601-624 (1993)]. For a codimension-two closed submanifold $$\Gamma$$ of $$H^{n + 1}$$, there are topological obstructions for $$\Gamma$$ to bound a hypersurface with $$K > -1$$ [the first author, Bull. Sci. Math., II. Sér. 117, No. 2, 211-239 (1993; Zbl 0787.53046)].

##### MSC:
 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
##### Keywords:
Gauss-Kronecker curvature; asymptotic boundary
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