Harmonic Maps and Minimal Surfaces into Spaces of Curvature Bounded from Above

Connecticut College, New London CT

Investigators

Abstract

DMS-0072483 Chikako Mese The principal investigator proposes to study geometric variational problems, with particular emphasis on harmonic maps in spaces with singularities. Motivated by questions in geometry and algebra, the study of harmonic maps to singular targets, particularly metric spaces of curvature bounded from above, has attracted the attention of many mathematicians. In particular, many remarkable results have been obtained by Gromov-Schoen, Korevaar-Schoen, and Jost. The PI proposes to develop the minimal surface theory as a continuation of the study of the harmonic map theory. The PI has shown that minimal surfaces in metric spaces of curvature bounded from above generalize several important properties of the classical minimal surfaces. We are interested in the development of a higher dimensional analogue of the theory advanced thus far. The PI will also investigate harmonic maps between surfaces when the target is given a metric of curvature bounded from above. We are particularly interested in understanding the behavior of harmonic maps when the target metric has singularities associated with curvature concentrations. Furthermore, we hope to gain an understanding of the behavior of harmonic maps when we vary the target metrics in a certain class. This in turn will be used to study Teichmuller spaces. The mathematical study of harmonic maps is natural and physically significant. This is because physics dictates that most natural actions occur in a way to minimize certain quantities. For instance, it is well known that the path of light in space is affected by gravity, and this path can be realized by a curve which minimizes arclength with respect to a certain metric. A thin film (such as a soap film formed on a closed wire frame) will come to a configuration which minimizes the surface area. Both these configurations can also be represented by energy minimizing maps. As critical points of the energy functional, harmonic maps are mathematical models of natural phenomena and provide an interesting mathematical study.