Analysis Aspects of Willmore Surfaces

Abstract: A new formulation for the Euler-Lagrange equation of the Will-more functional for immersed surfaces in Rm is given as a nonlinear elliptic equation in divergence form, with non-linearities comprising only Jacobians. Letting ~H be the mean curvature vector of the surface, our new formulation reads L ~H = 0, where L is a well-defined locally invertible self-adjoint elliptic op-erator. Several consequences are studied. In particular, the long standing open problem asking for a meaning to the Willmore Euler-Lagrange equation for im-mersions having only L2-bounded second fundamental form is now solved. The regularity of weak Willmore immersions with L2-bounded second fundamental form is also established. Its proof relies on the discovery of conservation laws which are preserved under weak convergence. A weak compactness result for Willmore surfaces with energy less than 8π (the Li-Yau condition ensuring the surface is embedded) is proved, via a point removability result established for Wilmore surfaces in Rm, thereby extending to arbitrary codimension the main result in [KS3]. Finally, from this point-removability result, the strong com-pactness of Willmore tori below the energy level 8π is proved both in dimension 3 (this had already been settled in [KS3]) and in dimension 4.