The impact of topology on variational problems between manifolds

Princeton University, Princeton NJ

Investigators

Abstract

Abstract Award: DMS-0101050 Principal Investigator: Fengbo Hang This project will study the impact of topology on some variational problems for maps between manifolds and investigate some conformal invariant differential equations. Sobolev spaces of weakly differentiable maps appear naturally in the solution of variational or differential equations problems, but the fundamental theorems on Sobolev spaces of maps between regions in Euclidean spaces are not necessarily reproduced for maps between manifolds. Topology in the domain and range of a mapping space can prevent approximation of an arbitrary Sobolev map by smooth maps and can force singularities to appear in solutions to variational problems. The principal investigator's work aims to understand the analytical description of some of these topologically forced singularities. Some particular topics include the rectifiability of defect measure of a general Sobolev map, Ginzburg-Landau type approximation for maps which are not weak limits of smooth maps, gap phenomena, extension of maps with energy control, travelling waves in planar antiferromagnets, and singularities of minimizing p-harmonic maps. Tools include geometric measure theory, analytical techniques for simplified Ginzburg-Landau models, differential topology, and homotopy theory. Physical problems often admit formulations that seek to minimize energy or some energy-like quantity over a large family of allowable configurations. For example, the path of a beam of light follows the shortest route between two points, minimizing time and distance. An important mathematical device for solving such "variational problems" is sometimes called the direct method: a sequence of realizable configurations that drive energy down toward a lowest possible value is used to construct a genuine minimizer, usually by picking out a subsequence that converges. To make this process work mathematicians tend to work in enriched environments such as the Sobolev spaces cited above. These make convergence easier to establish, and then one has to show that the candidate minimizer is actually present in the original setting. It is a relatively recent discovery that this procedure sometimes fails if the domain and range of the maps being studied has a bit of topological complexity, and the principal investigator's projects seek to analyze the nature of these occasional failures.