Bubble tree of branched conformal immersions and applications to the Willmore functional

Abstract

We develop a bubble tree construction and prove compactness results for $W^{2,2}$ branched conformal immersions of closed Riemann surfaces, with varying conformal structures whose limit may degenerate, in ${\Bbb R}^n$ with uniformly bounded areas and Willmore energies. The compactness property is applied to construct Willmore type surfaces in compact Riemannian manifolds. This includes (a) existence of a Willmore $2$-sphere in ${\Bbb S}^n$ with at least 2 nonremovable singular points (b) existence of minimizers of the Willmore functional with prescribed area in a compact manifold $N$ provided (i) the area is small when genus is $0$ and (ii) the area is close to that of the area minimizing surface of Schoen-Yau and Sacks-Uhlenbeck in the homotopy class of an incompressible map from a surface of positive genus to $N$ and $\pi_2(N)$ is trivial (c) existence of smooth minimizers of the Willmore functional if a Douglas type condition is satisfied.