Analysis aspects of Willmore surfaces

Abstract

A new formulation for the Euler–Lagrange equation of the Willmore functional for immersed surfaces in ℝ^m is given as a nonlinear elliptic equation in divergence form, with non-linearities comprising only Jacobians. Letting $\vec{H}$ be the mean curvature vector of the surface, our new formulation reads ${\mathcal{L}}\vec{H}=0$, where $\mathcal{L}$ is a well-defined locally invertible self-adjoint elliptic operator. Several consequences are studied. In particular, the long standing open problem asking for a meaning to the Willmore Euler–Lagrange equation for immersions having only L ²-bounded second fundamental form is now solved. The regularity of weak Willmore immersions with L ²-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 ℝ^m, thereby extending to arbitrary codimension the main result in [KS3]. Finally, from this point-removability result, the strong compactness 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.