Mathematical Sciences: Geometric Evolution of Curves, Surfaces, and Manifolds

小行星9626685 拟议的研究是在该地区的几何演化方程：曲线和曲面的曲率向量，共形流动的黎曼流形，和里奇流度量的演变是其中的主题要追求。研究者将研究里奇流的汉密尔顿的Harnack不等式和几何演化方程的其他Harnack不等式，以期了解里奇流的解在各种假设下的全局行为。 几何流在许多应用中出现。在计算机视觉中，曲率流已被用于平滑图像和增强边界（膨胀流最近在计算机视觉领域受到了极大的关注，其中小气泡被膨胀以检测图像的边界）;高斯曲率流模拟了水波对石头的磨损以及通过以随机角度撞击使滚珠轴承变圆的过程。此外，里奇流已被应用于分类问题的黎曼几何。

9626685 Chow The proposed research is in the area of geometric evolution equations: evolution of curves and surfaces by their curvature vectors, conformal flows of Riemannian manifolds, and the Ricci flow for metrics are among the topics to be pursued. The investigator will study Hamilton's Harnack inequality for the Ricci flow and other Harnack inequalities for geometric evolution equations with a view towards understanding the global behavior of solutions of Ricci flow under various assumptions on the geometry of the underlying manifold. Geometric flows arise in numerous applications. In computer vision, curvature flows have been used to smooth images and enhance boundaries (expanding flows have received a great deal attention lately in the field of computer vision, where small bubbles are expanded to detect boundaries of images); the Gauss curvature flow models the wearing of stones by water waves and the process of making ball-bearings round by impacting at random angles. Also, the Ricci flow has been applied to classification problems in Riemannian geometry.