*Accepted Paper*

**Inserted:** 12 jan 2012

**Last Updated:** 10 dec 2013

**Journal:** Math. Annalen

**Year:** 2011

**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 $f:S^2 \to M$ 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}

H

^2+1$, where $H$ is the mean curvature.

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