Bubble tree of branched conformal immersions and applications to the Willmore functional. (English) Zbl 1298.53007

Chen, Jingyi; Li, Yuxiang

Am. J. Math. 136, No. 4, 1107-1154 (2014).

Summary: 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 \(\mathbb R^n\) 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 \(\mathbb S^n\) 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.

MSC:

53A07	Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
49Q10	Optimization of shapes other than minimal surfaces

Keywords:

Willmore energies; 2 nonremovable singular points; Douglas type condition