My primary research interests lie in the field of global differential geometry and its interactions with topology, analysis, and complex geometry.
One of my research topics is the classification of isoparametric hypersurfaces in spheres, i.e., those hypersurfaces in spheres with constant principal curvatures. Using methods from algebraic topology Hans Friedrich Münzner proved that the number of distinct principal curvatures g can be only 1, 2, 3, 4, or 6. The possible multiplicities of the curvature distributions were classified by Uwe Abresch, Hans Friedrich Münzner and Stephan Stolz and coincide with the multiplicities in the known examples. While Elie Cartan classified the isoparametric hypersurfaces with g less or equal to 3, the classification in the cases g=4 and g=6 is still not complete.
Another topic I am interested in is equivariant differential geometry: geometric variational problems frequently lead to analytically extremely hard, non-linear partial differential equations, where the standard methods fail. Thus finding non-trivial solutions is challenging. The idea of equivariant differential geometry is to study solutions with a certain minimum level of symmetry (group actions with low cohomogeneity), and use the symmetry to reduce the original problem to systems of non-linear ordinary differential equations, typically with singular boundary values. Important examples include stable minimal cones, harmonic maps between Riemannian manifolds and isoperimetric domains. I am in particular interested in the construction of harmonic maps between Riemannian manifolds.
Recently I started to consider some problems in the context of Spectral Geometry.