B04 Harmonic maps and symmetry

The study of harmonic maps and their relatives is one of the most intensely studied research area in geometric analysis. It has found many applications within Mathematics; e.g. in algebraic geometry, where a specific version of the still-open Hodge conjecture was proven, or in Differential Geometry, where heat flow methods developed for harmonic map heat flow have been a source of inspiration for Richard Hamilton's early work on the Ricci flow. At the same time fundamental problems are still open, like the Chen conjecture or the existence of a biharmonic map from the two-torus to the two-sphere of degree one.

In this project we study harmonic maps and relatives, i.e. biharmonic maps and Yang-Mills fields, between Riemannian manifolds with symmetries and investigate the geometry of some of these manifolds themselves. More concretely, we plan to construct harmonic and biharmonic maps between singular codimension one metric foliations; to prove the existence of specific hairy wormholes and to classify isoparametric hypersurfaces in spheres.