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Some results on the qualitative behavior of solutions to the Ricci flow and other geometric evolution equations

  • Author(s): Kotschwar, Brett L.
  • et al.

In first part of this thesis we consider the Ricci flow, an evolution equation for Riemannian metrics introduced by Richard Hamilton. In dimensions two and three, the work of Hamilton and Perelman has effectively shown that the only gradient shrinking solitons are the round sphere, flat Euclidean space, the standard cylinder, and their quotients. Our first result is a classification of rotationally symmetric shrinking solitons in all dimensions, which we accomplish by the analysis of a certain system of ODE similar to one first considered by Bryant and Ivey for steady and expanding solitons. We also present an elementary proof of the uniqueness of a certain two-dimensional expanding soliton among those with positive curvature, which is an analog of a result of Chen, Lu, and Tian in the case of compact shrinking surface solitons. Next, we generalize an argument of Lees and Protter to prove a unique-continuation theorem for evolving tensor fields satisfying a certain parabolic differential inequality. As applications, we obtain unique -continuation theorems for solutions to the K"ahler- Ricci and Ricci-DeTurck flows, as well as a proof that a solution to the Ricci flow cannot become Einstein in finite time. In the next part, we consider differential Harnack inequalities for evolving convex hypersurfaces of the type proved by Hamilton, Chow, and Andrews. Modifying an approach of Chow, Chu, and Knopf, we exhibit a realization of the full Harnack quadratic as the second fundamental form of a certain degenerate immersion of the space-time track. By means of this realization, we provide a new geometric interpretation of Andrews's inequality in the case of isotropic flows and use the machinery to give a new proof of Hamilton's inequality for the mean curvature flow. We also show that Andrews's Gauss map technique can be used to obtain new Harnack inequalities for complete space-like surfaces in Minkowski space. Finally, via a Bernstein-type estimate and a maximum principle of Karp and Li, we extend a gradient estimate of Hamilton for the heat equation to complete manifolds. With this estimate, we obtain a sharp inequality for the heat kernel on complete manifolds of non-negative Ricci curvature

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