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

TL;DR

This paper proves the existence of immersed spheres that minimize specific curvature functionals in compact 3-manifolds under certain curvature conditions, advancing understanding of geometric variational problems.

Contribution

It establishes the existence of curvature-minimizing immersed spheres in 3-manifolds with positive or bounded sectional curvature, under particular scalar curvature assumptions.

Findings

Existence of spheres minimizing the L^{2} norm of the second fundamental form in positively curved manifolds.

Existence of spheres minimizing a combined curvature functional when scalar curvature exceeds 6 at some point.

Results depend on curvature bounds and scalar curvature conditions of the ambient manifold.

Abstract

We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold M. Under the assumption that the sectional curvature of M is strictly positive, we prove the existence of a smoothly immersed sphere minimizing the L^{2} integral of the second fundamental form. Assuming instead that the sectional curvature is less than or equal to 2, and that there exists a point in M with scalar curvature bigger than 6, we obtain a smooth 2-sphere minimizing the integral of 1/4|H|^{2} +1, where H is the mean curvature vector.