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

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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.