# Willmore Spheres in Compact Riemannian Manifolds

**published**
**year:** 2012

**journal:** Advances in Math.

**abstract:** The paper is devoted to the variational analysis of the Willmore, and other $L^2$ curvature functionals, for 2-d surfaces immersed in a compact riemannian $3\leq m$-manifold $(M^m,h)$; the double goal of the paper is on one hand to give the right setting for doing the calculus of variations (including min max methods) of such functionals for immersions into manifolds and on the other hand to prove existence of possibly branched Willmore spheres under curvature or topological conditions. For this purpose, using the integrability by compensation, we develop the regularity theory for the critical points of such functionals; a crucial step consists in writing the Euler-Lagrange equation (which is a system), first in a conservative form making sense for weak $W^{1,\infty}\cap W^{2,2}$ immersions, then as a system of conservation laws. Exploiting this new form of the equations we are able on one hand to prove full regularity of weak solutions to the Willmore equation in any codimension, on the other hand to prove a rigidity theorem concerning the relation between CMC and Willmore spheres. One of the main achievements of the paper is that for every non null 2-homotopy class $0\neq \gamma \in \pi_2(M^m)$ we produce a canonical 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 immersed spheres in homotopy class constructed in a celebrated paper by Sacks and Uhlembeck in situations when they do not exist); moreover for every ${\cal A}>0$ we minimize the Willmore functional among connected families of weak, possibly branched, immersions of $S^2$ having total area ${\cal A}$ and we prove full regularity for the minimizer. Finally, under a mild curvature condition on $(M^m,h)$, we minimize $\int(|\mathbb I|^2+1)$, where ${\mathbb I}$ is the second fundamental form, among weak possibly branched immersions of $S^2$ and we prove the regularity of the minimizer.

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