Differential geometry

In the field of Differential Geometry we are concerned with Riemannian manifolds or more generally (inner) metric spaces. We are interested in the interplay between their curvature and global topological and analytical properties, which in general are strongly intertwined. For instance, it is well known that certain positivity assumptions on the curvature imply topological obstructions of the underlying manifold. Curvature bounds also determine certain properties of elliptic operators. We also investigate group actions on such spaces.

We work on various rigidity and classification problems for Riemannian manifolds admitting isometric group actions. Assuming positive sectional curvature and (very weak) symmetry assumption we aim to obtain topological constraints for the underlying smooth manifold. Assuming the Einstein condition with negative Einstein constant and symmetry assumptions we aim for rigidity results (as in the homogeneous case). We are also interested in Riemannian manifolds all of whose geodesics are closed, in Alexandrov geometry, Cat(0) spaces, Lie groups (and locally compact groups) and geometric group theory.

In our work on geometric analysis, we focus on elliptic differential operators describing the Ricci or scalar curvature of a Kähler metric. The main theme  is the existence, asymptotic analysis and stability of solutions in relation to algebro-geometric properties of the underlying complex manifold. We are also interested  in area minimizing hypersurfaces, potential theory and Gromov hyperbolic spaces.

We work on Hamilton's Ricci flow, a (weakly) parabolic geometric evolution equation on the space of all Riemannian metrics. Ricci flow has been proved extremely successful in recent years for showing many deep theorems in Differential Geometry. Our focus lies on Ricci flow in higher dimensions, heat flow methods,  new Ricci flow invariant curvature conditions, the dynamical Alekseevskii conjecture and on Kähler Ricci flow.