Research Projects of Prof. Smoczyk
Geometric evolution equations have been applied to a variety of problems in differential geometry, topology, analysis and physics. Applications include the Penrose inequality, Poincaré and geometrization conjectures, uniformization theorems in Kähler geometry, classification of Riemannian and Kählerian manifolds, the 1/4-pinching theorem, the Schönfliess conjecture, Arnol'd conjecture, existence of canonical maps, connections, almost complex structures and metrics such as minimal and harmonic maps, Yang-Mills and Einstein connections, integrable almost complex structures, Einstein metrics and metrics of prescribed scalar or Ricci curvature.
The research interests of Professor Knut Smoczyk and his group mainly focus on geometric evolution equations. Mean curvature flows in higher codimension and Lagrangian mean curvature flows are of particular interest. Lagrangian mean curvature flows exist on Kähler-Einstein manifolds and more generally on almost Kähler manifolds equipped with an Einstein connection. This is the case for the cotangent bundle of a Riemannian manifold. In the Lagrangian context there exists a deep connection to mirror symmetry, T-duality, the Arnol'd conjecture and the topology of the group of symplectomorphisms of a Kähler-Einstein manifold.
Besides these parabolic methods his group has recently worked on corresponding elliptic problems, e.g. on the classification of minimal maps between Riemannian and Kählerian manifolds, special Lagrangian geometry and self-similar solutions of the mean curvature flow in higher codimension.