Some Variational Problems in Differential Geometry

9704482 Escobar This project lies in the area of Riemannian geometry. More specifically, the investigator is to study the total mean curvature plus the total scalar curvature functional, the Stekloff problem, isoperimetric inequality, and harmonic maps between noncompact manifolds. Also, the investigator has been active in developing collaborative research programs between the United States and Colombia, organizing various conferences and workshops. Riemannian manifolds are higher dimensional generalizations of curved surfaces. Such manifolds have well-known applications in theoretical physics. A Riemannian manifold possesses a notion of distance, or a metric. And by taking the derivative of the metric one can measure its curvature. Harmonic maps and harmonic functions constitute an important class of mappings between Riemannian manifolds. Harmonic maps have a variety of applications, e.g., they can be used to describe the temperature distribution on a homogeneous lamina.