Willmore spheres in compact Riemannian manifolds. (English) Zbl 1294.30046

Mondino, Andrea; Rivière, Tristan

Adv. Math. 232, No. 1, 608-676 (2013).

Summary: The paper is devoted to the variational analysis of the Willmore and other \(L^{2}\) curvature functionals, among immersions of 2-dimensional surfaces into a compact Riemannian \(m\)-manifold \((M^{m},h)\) with \(m>2\). The goal of the paper is two-fold, on one hand, we give the right setting for doing the calculus of variations (including minmax methods) of such functionals for immersions into manifolds and, on the other hand, we prove existence results for possibly branched Willmore spheres under various constraints (prescribed homotopy class, prescribed area) or under curvature assumptions for \(M^{m}\). To this aim, using the integrability by compensation theory, we first establish the regularity for the critical points of such functionals. We then prove a rigidity theorem concerning the relation between CMC and Willmore spheres. Then we prove that, for every non null 2-homotopy class, there exists a representative given by a Lipschitz map from the 2-sphere into \(M^{m}\) realizing a connected family of conformal smooth (possibly branched) area constrained Willmore spheres (as explained in the introduction, this comes as a natural extension of the minimal immersed spheres in homotopy class constructed by J. Sacks and K. Uhlenbeck [Ann. Math. (2) 113, 1–24 (1981; Zbl 0462.58014)], in situations when they do not exist). Moreover, for every \(A>0\) we minimize the Willmore functional among connected families of weak, possibly branched, immersions of the 2-sphere having prescribed total area equal to A and we prove full regularity for the minimizer. Finally, under a mild curvature condition on \((M^{m},h)\), we minimize the sum of the area with the square of the \(L^{2}\) norm of the second fundamental form, among weak possibly branched immersions of the two spheres and we prove the regularity of the minimizer.

MSC:

30C70	Extremal problems for conformal and quasiconformal mappings, variational methods
58E15	Variational problems concerning extremal problems in several variables; Yang-Mills functionals
58E30	Variational principles in infinite-dimensional spaces
49Q10	Optimization of shapes other than minimal surfaces
53A30	Conformal differential geometry (MSC2010)
35R01	PDEs on manifolds
35J35	Variational methods for higher-order elliptic equations
35J48	Higher-order elliptic systems
35J50	Variational methods for elliptic systems

Keywords:

conformal differential geometry; weak immersions; minimal surfaces; optimal surfaces; geometric PDE; elliptic regularity theory

Citations:

Zbl 0462.58014