Calculus of Variations and Geometric Measure Theory

E. Kuwert - A. Mondino - J. Schygulla

Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds

created by mondino on 12 Jan 2012
modified on 10 Dec 2013


Accepted Paper

Inserted: 12 jan 2012
Last Updated: 10 dec 2013

Journal: Math. Annalen
Year: 2011


We study curvature functionals for immersed $2$-spheres in a compact, three-dimensional Riemannian manifold $M$. Under the assumption that the sectional curvature $K^M$ is strictly positive, we prove the existence of a smooth immersion $f:S^2 \to M$ minimizing the $L^2$ integral of the second fundamental form. Assuming instead that $K^M \leq 2$ and that there is some point $\overline{x} \in M$ with scalar curvature $R^M(\overline{x}) > 6$, we obtain a smooth minimizer $f:S^2 \to M$ for the functional $\int \frac{1}{4}
^2+1$, where $H$ is the mean curvature.