The Pontrjagin-Hopf invariants for Sobolev maps

Variational problems with topological constraints can exhibit many interesting mathematical properties. There are many open mathematical questions in this area. The diverse properties these problems possess give them the potential for different applications. We find variational problems with topological constraints in high energy physics, hydrodynamics, and material science, [9, 3, 48]. Usually, the functional to be minimized represents energy and the topological constraint is that the map must be in a fixed homotopy class. An energy functional dictates a natural class of maps – the finite energy maps. For some models finite energy maps include discontinuous maps and this introduces new technical difficulties into the homotopy theory. Many traditional arguments have been designed for continuous or smoother maps and do not work for finite energy maps. Thus, one has to look for new approaches and new interpretations that survive the lack of regularity. This may lead to new geometric results and require more subtle analytic techniques. Even if all finite energy maps were continuous, one would want analytic expressions for complete homotopy invariants in order to apply the direct method in the calculus of variations. This is another challenge in geometry. This is a relatively new area of research with interesting interplay between geometry and analysis. The first steps have been made by studying the homotopy classes for Sobolev maps, see [17, 49, 10, 11, 15, 14, 27]. One can view Sobolev maps as the maps with finite Sobolev energy aka the Sobolev norm. These studies are very important because Sobolev spaces are a basic technical tool in analysis.