Gradient Flow for the Willmore Functional in Riemannian Manifolds of bounded Geometry

Abstract

We consider the $L^2$ gradient flow for the Willmore functional in Riemannian manifolds of bounded geometry. In the euclidean case E.\;Kuwert and R.\;Sch\"atzle [\textsl{Gradient flow for the Willmore functional,} Comm. Anal. Geom., 10: 307-339, 2002] established a lower bound of a smooth solution of such a flow, which depends only on how much the curvature of the initial surface is concentrated in space. In a second joint work [\textsl{The Willmore flow with small initial energy,} J.\;Differential Geom., 57: 409-441, 2001] the aforementioned authors proved that a suitable blow-up converges to a nonumbilic (compact or noncompact) Willmore surface. In the lecture notes of the first author [\textsl{The Willmore Functional,} unpublished lecture notes, ETH Z\"urich, 2007] the blow-up analysis was refined. In the present work we intend to generalize the results mentioned above to the Riemannian setting.