Harnack inequalities for curvature flows and applications

该项目的研究对象是黎曼流形的严格凸超曲面在曲率变形下的行为，例如在平均曲率流及其完全非线性变体下。我们的目标是推导出Harnack不等式的流速，一个问题，主要是在欧几里得空间。作为应用程序，我们的目标是新的收敛结果，在球体中的流动，特别是古代的解决方案的分类和适当重新缩放超曲面的光滑收敛。这些结果应证明为一类流速，其中包括幂的高斯曲率，并应处理在流形的非恒定截面曲率。获得一个很好的控制的流速通常是一个主要的困难，在这样的问题，这将克服与Harnack不等式。为了在一般情况下证明它们，我们的目标是找到一个非常合适的替代品，著名的高斯参数化的严格凸超曲面在欧几里得空间。这些问题与当前的研究状况相联系，并突出了曲率流理论中一些未回答的问题。

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.