01126nas a2200169 4500008004100000022001400041245009000055210006900145260000800214300001400222490000800236520060400244100001800848700002000866700002400886856004600910 2014 eng d a1432-180700aExistence of immersed spheres minimizing curvature functionals in compact 3-manifolds0 aExistence of immersed spheres minimizing curvature functionals i cJun a379–4250 v3593 a
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.
1 aKuwert, Ernst1 aMondino, Andrea1 aSchygulla, Johannes uhttps://doi.org/10.1007/s00208-013-1005-3