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

TitleExistence of immersed spheres minimizing curvature functionals in compact 3-manifolds
Publication TypeJournal Article
Year of Publication2014
AuthorsKuwert, E, Mondino, A, Schygulla, J
JournalMathematische Annalen
Volume359
Pagination379–425
Date PublishedJun
ISSN1432-1807
Abstract

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.

URLhttps://doi.org/10.1007/s00208-013-1005-3
DOI10.1007/s00208-013-1005-3

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