Analysis aspects of Willmore surfaces

Abstract

We found a new formulation to the Euler-Lagrange equation of the Willmore functional for immersed surfaces in ${\R}^m$. This new formulation of Willmore equation appears to be of divergence form, moreover, the non-linearities are made of jacobians. Additionally to that, if $\bH$ denotes the mean curvature vector of the surface, this new form writes ${\mathcal L}\bH=0$ where ${\mathcal L}$ is a well defined locally invertible self-adjoint operator. These 3 facts have numerous consequences in the analysis of Willmore surfaces. One first consequence is that the long standing open problem to give a meaning to the Willmore Euler-Lagrange equation for immersions having only $L^2$ bounded second fundamental form is now solved. We then establish the regularity of weak $W^{2,p}-$Willmore surfaces for any $p$ for which the Gauss map is continuous : $p>2$. This is based on the proof of an $\epsilon-$regularity result for weak Willmore surfaces. We establish then a weak compactness result for Willmore surfaces of energy less than $8\pi-\delta$ for every $\delta>0$. This theorem is based on a point removability result we prove for Wilmore surfaces in ${\R}^m$. This result extends to arbitrary codimension a result that E.Kuwert and R.Schaetzle established for surfaces in ${\R}^3$. Finally, we deduce from this point removability result the strong compactness, modulo the M\"obius group action, of Willmore tori below the energy level $8\pi-\delta$ in dimensions 3 and 4. The dimension 3 case was already solved in a previous work.