The projects in this proposal aim to advance existing and to create new approaches to attack existence problems of prescribed curvature type that are of interest in geometry and mathematical physics. The P.I. proposes to refine techniques from his previous work to address existence, regularity, and global properties of apparent and generalized apparent horizons in general relativity. In particular, the P.I. plans to investigate these quasi-local notions of the event horizon in light of certain lower order features, much akin to those of minimal and constant mean curvature surfaces, for their relation to positive mass type theorems, and in particular to the space time Penrose conjecture. In a separate project, the P.I. plans to study the extent to which these techniques can be adapted to include an existence theory for a wide range of embedded hypersurfaces of prescribed curvature. A second, independent direction in the proposed research is a potentially open-ended collaboration with N. Kapouleas on gluing problems for constant mean curvature surfaces.
The projects in this proposal are inspired and driven by allowing analytical, geometric, and physical intuition to guide one another in the spirit of R. Schoen and S.-T. Yau. The special surfaces whose existence and properties will be investigated here are relevant in a wide variety of geometric and physical contexts, in particular in mathematical relativity - Einstein's theory of gravitation. A major objective of the proposed research is to exploit a parallel between well-known properties of soap films on the one hand and the geometry of black hole boundaries on the other hand. Through this connection, powerful techniques from analysis and geometry have entered the picture and are now at hand to further our understanding of the geometry of black holes.
