Bubble tree of a class of conformal mappings and applications to Willmore functional

Details Bubble tree of a class of conformal mappings and applications to Willmore functional Bubble tree of a class of conformal mappings and applications to Willmore functional

2011-12-08

arxiv.org/abs/1112.1818v1

Mathematics

Differential Geometry

38 pages, 4 figures

Scientific paper

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 a compact Riemannian manifold 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 ${\mathbb S}^n$ 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.

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