Minimal submanifolds in riemannian geometry

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This research proposal deals with applications of analytic methods of minimal surface theory to geometric and topological problems. Minimization problems arise naturally in many branches of mathematics and science. For example, problems in navigation involve finding paths of least length (`geodesics') on the earth's surface. Minimal surfaces, which are two-dimensional analogs of geodesics, are minimizers (or simply critical points) of the area function, and arise naturally in material science; for example in fluid interface problems and elasticity problems. A simple physical example of a minimal surface is the soap film that forms after dipping a wire frame into a soap solution. By the laws of surface tension this soap film has the property that it is stable, that is it becomes larger under slight deformations. In modern geometry, minimal surfaces (and submanifolds) have had striking applications, for example to general relativity and low dimensional topology. The objects under investigation in this proposal are critical points for geometric variational problems. Such objects can often, under certain geometric conditions (such as a curvature condition) be shown to satisfy very special properties. This information can then be used to obtain information about the whole geometric class under consideration. The results of this project are aimed at increasing our understanding of the fundamental relationships between the curvature and topology of spaces.