TY - JOUR T1 - Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds JF - Mathematische Annalen Y1 - 2014 A1 - Kuwert, Ernst A1 - Andrea Mondino A1 - Johannes Schygulla AB -

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 $f:{\mathbb{S}}^2 \rightarrow M$ minimizing the $L^2$ integral of the second fundamental form. Assuming instead that $K^M \leq 2 $ and that there is some point $\bar{x}\in M$ with scalar curvature $R^M(\bar{x})>6$, we obtain a smooth minimizer $f:{\mathbb{S}}^2 \rightarrow M$ for the functional $\int \frac{1}{4}|H|^2+1$, where $H$ is the mean curvature.

VL - 359 UR - https://doi.org/10.1007/s00208-013-1005-3 ER -