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

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.

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
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