The existence of hypersurfaces of constant Gauss curvature with prescribed boundary. (English) Zbl 1070.58013

Guan, Bo; Spruck, Joel

J. Differ. Geom. 62, No. 2, 259-287 (2002).

The authors establish the existence of hypersurfaces of constant Gauss-Kronecker curvature with prescribed boundary. This is achieved by means of Monge-Ampere type equations. Important local properties of locally convex hypersurfaces with boundary are proven. A special attention is paid to the regularity of the constructed hypersurfaces.

Reviewer: Dumitru Motreanu (Perpignan)

MSC:

58E12	Variational problems concerning minimal surfaces (problems in two independent variables)
35J25	Boundary value problems for second-order elliptic equations
35J60	Nonlinear elliptic equations
53A10	Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Keywords:

hypersurfaces; curvature; Monge-Ampere equations; regularity