Coping with 'bubbling' and 'topological degeneration' in the Calculus of Variations

Already Jesse Douglas was aware of the fact that minimizing sequences for Dirichlet's integral of annulus-type surfaces spanning two parallel, co-axial planar circles might degenerate into a pair of discs. The characterization of "bubbling" of (approximate) harmonic maps from a closed surface to a closed Riemannian target manifold allowed Sacks-Uhlenbeck to conclude the existence of harmonic representatives for every homotopy class of maps in the case of target manifolds whose second fundamental group was trivial. Chang-Yang were able to give sufficient conditions for solving Nirenberg's problem for conformal metrics of prescribed Gauss curvature on the 2-sphere by studying the contribution of degenerate conformal metrics to the topological degree of the associated variational problem.

Inspite of these achievements, there still are many open questions related to the possible topological degeneration of comparison maps or "bubbling" in geometric variational problems, and we will discuss some of these questions.