Regulary and stability of curvature flows and their applications to geometric variational problems

• Applicant: Professor Dr. Felix Schulze
• Subject Area: Mathematics
• Term: from 2007 to 2014
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 62175069

• Final Report Year: 2017

Final Report Abstract

The project investigates the regularity and stability properties of curvature flows and their applications to geometric variational problems. We have obtained results about the finer singularity structure of mean curvature flow and shown a short-time existence result for the planar network flow from non-regular initial networks. This has direct applications in defining an evolution through singularities. Furthermore, we have proven a local regularity and a shorttime existence result for mean curvature flow with triple edges, which is the first step in understanding this natural extension of classical mean curvature flow. For Ricci flow we have obtained a stability result for hyperbolic space and understood the long-time behaviour of solutions from non-negatively curved manifolds with non-vanishing asymptotic volume ratio. For initial manifolds which have isolated conical singularities, modelled on non-negatively curved cones, we have shown that there exists a solution, starting from such initial data, which is immediately smooth. This is expected to have applications to understand a possible evolution through singularities in higher dimensions. Additionally we have obtained estimated on the singular set in Type I Ricci flows. For the problem of understanding the interaction of the Willmore functional with ambient curvature, we have shown that in an exterior region of an asymptotically Schwarzschild 3-manifold, there exists a foliation by spherical surfaces which are critical points of the Willmore functional under an area constraint. This has a direct connection with the concept of Hawking mass in General Relativity and thus provides a further natural coordinate system at infinity for an isolated gravitating system. We have furthermore investigated the long-time behaviour of area preserving Willmore flow in such an exterior region.

Publications

• Foliations of asymptotically flat spacetimes by surfaces of Willmore type, Math. Ann. 350, No. 1, 1–78 (2011)
  T. Lamm, J. Metzger and F. Schulze
  
• Stability of Hyperbolic space under Ricci-flow, Comm. Anal. Geom. 19, No. 5, 1023–1047 (2011)
  O.C. Schnürer, F. Schulze and M. Simon
  
• Stabilität translatierender Lösungen des graphischen Mittleren Krümmungsflusses unter unbeschränkten Störungen, PhD-thesis, Free University Berlin (2012)
  A. Hammerschmidt
  
• Expanding solitons with non-negative curvature operator coming out of cones, Math. Z. 275 (1-2), 625–639 (2013)
  F. Schulze and M. Simon
  
• Flächeninhaltserhaltender Willmore-Fluss im asymptotisch Schwarzschildschen, PhD-thesis, Free University Berlin (2014)
  F. Jachan
  
• On Short Time Existence for the planar network flow
  T. Ilmanen, A. Neves and F. Schulze
  
• Uniqueness of compact tangent flows in Mean Curvature Flow, J. Reine Angew. Math. (690), 163–172 (2014)
  F. Schulze
  (See online at https://doi.org/10.1515/crelle-2012-0070)
• The size of the singular set of a Type I Ricci flow, (2015), J. Geom. Anal.
  P. Gianniotis
  
• A local regularity theorem for mean curvature flow with triple edges
  F. Schulze and B. White
  
• Ricci flow from spaces with isolated conical singularities
  P. Gianniotis and F. Schulze