Mean curvature flow in higher co-dimensions

• Applicant: Professor Dr. Knut Smoczyk
• Subject Area: Mathematics
• Term: from 2015 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 274149541

• Final Report Year: 2019

Final Report Abstract

In the project we obtained the following major results: 1. We derived a complete description of the evolution of an area decreasing map f : M → N , induced by the mean curvature of their graph, in the situation where M and N are complete Riemann surfaces with bounded geometry, M being compact, for which their sectional curvatures σM and σN satisfy min σM ≥ sup σN . Moreover, several decay estimates for the second fundamental form and the mean curvature were obtained. 2. Properly embedded connected translating surfaces in R3 that are C1-asymptotic to two half-planes were studied and it was shown that either they coincide with half-planes or grim reaper cylinders. 3. The Lagrangian mean curvature flow of almost calibrated Lagrangian submanifolds in Calabi-Yau manifolds was studied and it was shown that there exists an optimal control on the evolving measure. This could be used to classify certain type-II singularities as the products of the grim reaper Γ with minimal Lagrangian submanifolds M ⊂ C^m−1 . 4. The evolution of spheres, in particular of the Whitney sphere, in euclidean space under the Lagrangian mean curvature flow was studied and under a natural non-negativity condition on the Ricci curvature it was shown that the developing type-II singularities must be grim reaper cylinders.

Publications

• Mean curvature flow of area decreasing maps between Riemann surfaces, Annals of Global Analysis and Geometry, 53 (2018), no. 1, 11–37
  A. Savas-Halilaj, K. Smoczyk
  (See online at https://doi.org/10.1007/s10455-017-9566-0)
• A characterization of the grim reaper cylinder, Journal für die reine und angewandte Mathematik, 746 (2019), 209–234
  F. Martin, J. Perez-Garcia, A. Savas-Halilaj, K. Smoczyk
  (See online at https://doi.org/10.1515/crelle-2016-0011)
• Lagrangian mean curvature flow of Whitney spheres, in: Geometry & Topology
  A. Savas-Halilaj, K. Smoczyk
  (See online at https://doi.org/10.2140/gt.2019.23.1057)
• Local non-collapsing of volume for the Lagrangian mean curvature flow, Calc. Var. and PDE 58 (2019), no.1, 58:20
  K. Smoczyk
  (See online at https://doi.org/10.1007/s00526-018-1458-z)