Starshaped locally convex hypersurfaces with prescribed curvature and boundary

Abstract

In this paper we find strictly locally convex hypersurfaces in ${\mathbb{R}}^{n+1}$ with prescribed curvature and boundary. The main result is that if the given data admits a strictly locally convex radial graph as a subsolution, we can find a radial graph realizing the prescribed curvature and boundary. As an application we show any smooth domain on the boundary of a compact strictly convex body can be deformed to a smooth hypersurface with the same boundary (inside the convex body) and realizing any prescribed curvature function smaller than the curvature of the body.

24 pages. References updated. Published in The Journal of Geometric Analysis

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• Lipschitz Continuous Hypersurfaces with Prescribed Curvature and Asymptotic Boundary in Hyperbolic Space
• Strictly Locally Convex Radial Graphs of Prescribed Curvature and Boundary in Space Forms