UC Irvine

In this dissertation, we discuss the behavior of Willmore flow, a fourth-order geometric flow, for complete, properly immersed surfaces in Euclidean space.We develop a-priori estimates for weighted Willmore flows. The estimates are later used to generalize Kuwert and Schäzle’s short-time existence theorem in their 2002 article for complete surfaces with bounded geometry, to find a condition for uniqueness of Willmore flows on complete surfaces, and show gap phenomena of Willmore energy. We also discuss blowups of Willmore flow, as constructed by Kuwert and Schätzle in their 2001 article. We also discuss the Fredholm property of Laplacian operator on the space of normal vector fields, in view of weighted Sobolev spaces as defined by Lockhart in his 1987 article. We give a few conjectures regarding the Fredholm property of linearization of Willmore tensor, Łojasiewicz–Simon inequality, and stability of minimal surfaces with finite energy as Willmore surfaces.