Embeddedness of least area minimal hypersurfaces

Abstract

E. Calabi and J. Cao showed that a closed geodesic of least length in a two-sphere with nonnegative curvature is always simple. Using min-max theory, we prove that for some higher dimensions, this result holds without assumptions on the curvature. More precisely, in a closed $(n+1)$-manifold with $2\le n\le 6$, a least area closed minimal hypersurface exists and any such hypersurface is embedded. As an application, we give a short proof of the fact that if a closed three-manifold $M$ has scalar curvature at least $6$ and is not isometric to the round three-sphere, then $M$ contains an embedded closed minimal surface of area less than $4\pi$. This confirms a conjecture of F. C. Marques and A. Neves.

arXiv commentAdded condition (LS), revised Lemma 17, Proposition 18 and proof of Claim 1, results unchanged, added 1 picture. Added Section 3

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