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

2011/12/08 by Jingyi Chen, Yuxiang Li, Chen, Jingyi +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG

paper · pdf · doi:10.48550/arxiv.1112.1818

38 pages, 4 figures

arxiv created 2011/12/08 · arxiv updated 2011/12/09

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 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 Sⁿ 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 π₂(N) is trivial (c) existence of smooth minimizers of the Willmore functional if a Douglas type condition is satisfied.

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