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.

What is ... ?

• Obstructions to positive curvature

  There are only a few known examples of closed smooth manifolds admitting metrics of positive sectional curvature including spheres, real, complex and quaternionic projective spaces. An interesting question now is whether the list of examples is complete. To show this one needs results on topological invariants for Riemannian manifolds M with positive sectional curvature, i.e. obstructions to positive curvature. For example it is known that  the fundamental group of M is finite (Bonnet-Myers). Furthermore the sum of Betti numbers of M is bounded above by a constant that only depends on the dimension of M (Gromov). A conjecture due to Hopf from the 1930s says that the Euler characteristic of M is positive if the dimension of M is even. Kennard, Wiemeler and Wilking proved this conjecture under the addional assumption that the isometry group of M is large enough. Moreover, under further extra topological assumptions they could also compute the rational cohomology ring of M.

• Ricci flow

  Hamilton's Ricci flow is a (weakly parabolic) geometric evolution equation on the space of Riemannian metrics of a smooth manifold. It has been used to prove famous results in geometry and topology, like the differentiable sphere theorem by Brendle and Schoen in 2009 or the Poincare' conjecture by Perelman in 2003. Since Ricci flow behaves in a certain sense like a (non-linear) heat equation, a  maximum principles is known to exist. This allows one to obtain qualitative results for the Ricci flow by showing qualitative results for the associated Ricci flow ODE. Since this ODE has a strong Lie theoretic flavor, algebraic estimates have  been used to obtain convergence results for the normalized Ricci flow. This is also crucial for describing new Ricci flow invariant curvature conditions (in higher dimensions). We also study the Ricci flow on (Ricci flow invariant) subset of Riemannian metrics, like Kähler Ricci and  Ricci flow under symmetry assumptions.

• Einstein manifolds

  A Riemannian manifold (M n , g) is called Einstein if it has constant Ricci tensor, that is if ric(g) = λ · g, λ ∈ R. Although Einstein metrics on compact manifolds can be characterized as the critical points of the Einstein-Hilbert action, general existence and non-existence results are hard to obtain and there is no clear conjecture. Still there exist many examples of Einstein manifolds which have been constructed using bundle, symmetry and holonomy assumptions. Among many others, we mention here only Sasakian-Einstein metrics, Einstein metrics on spheres and Ricci-flat manifolds with holonomy G2 and Spin(7). Outstanding achievements within the last decade are the classification of compact Kähler-Einstein manifolds, the classification of non-compact homogeneous Einstein manifolds and gluing constructions for K\"ahler-Einstein manifolds.

• CAT(0) spaces

  A geodesic metric space is called a CAT(0) space if geodesic triangles are not thicker than triangles in euclidean space. Key examples are Riemannian symmetric spaces of noncompact type and metric cell complexes associated to Coxeter groups and buildings. We study actions of discrete groups, Lie groups and locally compact groups on such spaces, and the interplay between the group structure and the geometry of the space.

• Kähler manifolds

  Kähler manifolds are manifolds with three mutually compatible structures: a Riemannian, a complex and a symplectic structure. Examples of such manifolds are Riemann surfaces, ${C}^n$, ${C}P^n$ and their complex submanifolds.

  When one seeks metrics with prescribed Ricci curvature on a manifold (for example, solutions to the Einstein equations), the class of Kähler manifolds is especially well suited: a Kähler metric can be defined by a single smooth function, and prescribing the Ricci curvature amounts to imposing a single differential equation on this function. This is the so-called complex Monge-Ampère equation, a fully nonlinear elliptic PDE.

  In 1978 Yau achieved a breakthrough in analysis by solving the complex Monge-Ampère equation on compact Kähler manifolds. The study of the geometry of Yau's solutions continues to pose many challenging problems to this day.