Variational Problems in Riemannian Geometry

In differential geometry, many of the particularly interesting maps and geometric structures arise in the study of extrema of variational problems. Solutions of such a problem, in general, satisfy an elliptic partial differential equation and can often be obtained by deformation under a geometric flow. Particular examples include harmonic maps, Willmore surfaces and metrics with constant scalar curvature (in a given conformal class). These are critical points of, respectively, the energy, the Willmore and the Yamabe functional. Many other geometric problems, defined originally in some other way, turn out to admit a variational formulation, which then sheds new light on the question. This occurred, for example, with such highly relevant topics as manifolds with $G_2$ holonomy and with special Lagrangian submanifolds. Hence, geometric variational problems interact with many other areas of mathematics, and have strong relevance to integrable systems, mathematical physics and PDE.