Mathematical Sciences: Theoretical and Numerical Existence of Many Minimal Surfaces on Riemannian Manifolds

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The principal investigator will study minimal hypersurfaces in Riemannian manifolds. He will continue to develop and apply techniques from geometric measure theory and the geometric calculus of variations to existence problems for minimal surfaces. In addition the principal investigator will continue to develop numerical methods for constructing minimizers. The research supported by this award will attempt to describe how minimal surfaces lie in curved space. Minimal surfaces are surfaces which have minimal area for any prescribed boundary curve drawn on the surface. These surfaces frequently occur in nature because many processes develop according to some minimum energy principle. An important example is the solid- liquid boundary in a solidification process. Because there are so many important examples of geometric minimization in the physical sciences, a thorough understanding of the theoretical aspects of the process can be expected to have a broad impact on science.