On the existence of convex hypersurfaces of constant Gauss curvature in hyperbolic space. (English) Zbl 0823.53047

Rosenberg, Harold; Spruck, Joel

J. Differ. Geom. 40, No. 2, 379-409 (1994).

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)].

Reviewer: M.Craioveanu (Timişoara)

MSC:

53C42

Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Keywords:

Gauss-Kronecker curvature; asymptotic boundary