Variational problems for maps between manifolds

In order to understand the geometry of manifolds and the maps between them, it is often helpful to study specific maps that arise as the solutions of certain variational problems. This is one reason why harmonic maps, which are the critical points of the Dirichlet functional, have been studied extensively. This specific problem is very well understood now, but there are several variants of the problem that still give rise to open problems of a quite fundamental nature. These include the harmonic map equation with an extra term or biharmonic/polyharmonic maps. The purpose of this project is to study the regularity of these and the existence under certain boundary conditions or other side conditions.

For the regularity question, previous work for similar problems makes use of a monotonicity formula, which will need to be extended to the problems studied. The next step will most likely be based on ideas of Riviere, who rewrote similar equations with the help of suitable gauge transformations, so that they permit estimates in certain function spaces. For the problems studied in this project, this method will require some extensions, too.


Existence questions are interesting above all for biharmonic/polyharmonic maps. The underlying difficulty is a lack of coercivity of the energy functional due to possible energy concentration. Progress will require a detailed analysis of this concentration here.