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

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In this project we study regularity and stability properties of curvature flows and their applications to geometric variational problems. We are especially interested in the regularity theory for weak mean curvature flow in the formulation of the level-set-flow or Brakke-flow. We also want to study the dynamic stability of non-compact manifolds under curvature flows like mean curvature flow and Ricci flow. We hope to understand the long-time behavior of these flows near eternal solutions. Furthermore we aim at applying curvature flows of this type to geometric variational problems like isoperimetric inequalities and foliations of asymptotically flat space-times.