Curvature flows and convex geometry

该程序的目的是研究曲率流在凸和仿射几何中的应用。它的主要部分是通过高斯曲率的某些幂讨论凸超曲面的流动。从本质上讲，这些流将被研究，以了解一个超曲面如何随时间演变成一个更优的形状，并获得关于这些渐近形状的重要信息。在Brunn-Minkowski-Firey凸体理论中出现的一个著名问题推广了寻找具有规定曲率函数的凸体的经典Minkowski问题。在经典问题的扩展中，寻求具有规定L_p曲率函数的凸体，这一问题由于在信息论等其他领域的进一步应用而引起了人们的广泛关注。我们过去在广义问题上所做的工作已经通过曲率流的渐近行为得到了这种凸体的存在性和唯一性的结果。关键任务之一是为上述流程改进此方法，以解决一类其他更困难的情况。在这个项目中还有许多其他的几何问题。他们主要集中在两个方向：研究凸体的新仿射不变量，以及在欧几里得平面或空间中找到两个有界凸域的充分条件，使其中一个域包含另一个域的同余副本。尽管它们明显的多样性，我们的目标有一个共同的线索，即使用几何演化方程。来自偏微分方程、凸几何和微分几何的各种技术对于处理这些主题是必要的，每个技术本身都是该领域中具有挑战性和重要的问题。

The purpose of this program is to study applications of curvature flows to convex and affine geometry. Its main part concerns flows of convex hypersurfaces by certain powers of the Gauss curvature. In essence, these flows will be studied to understand how a hypersurface evolves in time to a more optimal shape and to get significant information on these asymptotic shapes.One of the famous questions that arises within the Brunn-Minkowski-Firey theory of convex bodies generalizes the classical Minkowski problem of finding convex bodies with prescribed curvature function. In the extension of the classical problem, one seeks convex bodies with prescribed L_p curvature function, a question which generated a lot attention in view of its further applications in other areas like information theory. Past work that we did on a case of the generalized problem has led to results on the existence and uniqueness of such convex bodies via the asymptotic behavior of a curvature flow. One of the key tasks is to refine this method for the aforementioned flows to solve a class of other, more difficult, cases.There are also a number of other geometric questions in the project. They concentrate along two main directions: the study of new affine invariants of convex bodies and the question of finding sufficient conditions for two bounded convex domains in the Euclidean plane or space such that one domain contains a congruent copy of the other.Despite their apparent diversity, our objectives have as a common thread the use of geometric evolution equations. A variety of techniques coming from PDE, convex and differential geometry are necessary to approach these topics, each in itself a challenging and important problem in the field.