Nonlinear Problems in Geometry

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We propose to study questions of existence of hypersurfaces of constant and prescribed curvature in hyperbolic space (with a given finite or asymptotic boundary) and the corresponding problems for spacelike hypersurfaces in Minkowski space. Among the many technical challenges involved is the need for a general maximum principle for the maximum principle curvature. We also intend to study models of black holes and cosmological dust where the choice of a suitable metric reduces the Einstein-Maxwell equations to a single semilinear elliptic equation. In addition to many subtle existence problems, the analysis of the geometric properties of the resulting spacetime leads to many interesting questions.Many diverse areas of interest, such as image processing and medical imaging , optimal design, computational biology and cosmology utilize models that either explicitly or implicitly involve the nonlinear elliptic equations that describe curvature or curvature flows (and even more complicated processes). For example, twenty five years ago the novel use of the level set mean curvature flow as a computational tool in image processing required a better theoretical understanding which led to new mathematical breakthroughs. Today, the underlying models and problems are much more sophisticated and underscore the need for a better theoretical understanding of the mathematics involved.