2023 Theses Doctoral

Ricci flow and positivity of curvature on manifolds with boundary

Chow, Tsz Kiu Aaron

In this thesis, we explore short time existence and uniqueness of solutions to the Ricci flow on manifolds with boundary, as well as the preservation of natural curvature positivity conditions along the flow.

In chapter 2, we establish the existence and uniqueness for linear parabolic systems on vector bundles for Hölder continuous initial data. We introduce appropriate weighted parabolic Hölder spaces to study the existence and uniqueness problem. Having developed the linear theory, we apply it to establish the existence and uniqueness for the Ricci-DeTurck flow, the harmonic map heat flow, and the Ricci flow with Hölder continuous initial data in Chapter 3.

In chapter 4, we discuss a general preservation result concerning the preservation of various curvature conditions during boundary deformation. Using a perturbation argument, we construct a family of metrics which interpolate between two metrics that agree on the boundary, and such family of metrics preserves various natural curvature conditions under suitable assumptions on the boundary data.

The results from chapters 2 through 4 will be utilized in proving the Main Theorems in chapter 5. In particular, we construct canonical solutions to the Ricci flow on manifolds with boundary from canonical solutions to the Ricci flow on closed manifolds with Hölder continuous initial data via doubling.