Starshaped locally convex hypersurfaces with prescribed curvature and boundary

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Johns Hopkins University
This dissertation is a record of the paper [19] by the author. We study the problem to find strictly locally convex hypersurfaces in 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 if a disjoint collection of smooth, closed, embedded, codimension 2 submanifolds in R^{n+1} can bound a convex hypersurface, which is a subsolution for the given curvature data and lies strictly on one side of every tangent hyperplane at the boundary, it can also bound a convex hypersurface realizing the given curvature data. We also 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. Readers: Professor Joel Spruck (Advisor) and Professor Jacob Bernstein
Strictly locally convex hypersurface, Radial graph, General curvature, Fully nonlinear elliptic partial differential equation