Harnack inequalities for curvature flows and applications

点击翻译按钮获取中文摘要

The object of investigation of this project is the behaviour of strictly convex hypersurfaces of Riemannian manifolds under deformation by their curvature, e.g. under the mean curvature flow and its fully nonlinear variants. We aim to derive Harnack inequalities for the flow speed, a problem which has been addressed mostly in the Euclidean space. As applications we aim for new convergence results for flows in the sphere, especially for the classification of ancient solutions and for smooth convergence of suitably rescaled hypersurfaces. These results shall be proven for a class of flow speeds which includes powers of the Gaussian curvature and shall also be treated in manifolds of non-constant sectional curvature. Obtaining a good control of the flow speed is usually a major difficulty in such problems, which shall be overcome with the Harnack inequalities. In order to prove them in a general setting, we aim to find a well-suited substitute for the well-known Gauss parametrization of a strictly convex hypersurface in the Euclidean space. These questions tie in with the current state of research and highlight some unanswered aspects in the theory of curvature flows.