Embeddedness of least area minimal hypersurfaces

TL;DR

This paper proves that in certain higher-dimensional manifolds, least area minimal hypersurfaces are embedded, extending classical results and confirming a conjecture about minimal surfaces in three-manifolds with scalar curvature bounds.

Contribution

It establishes the existence and embeddedness of least area minimal hypersurfaces in higher dimensions without curvature assumptions, and provides a new proof of a conjecture relating scalar curvature to minimal surfaces.

Findings

Least area minimal hypersurfaces are embedded in dimensions 2 to 6.

In three-manifolds with scalar curvature ≥ 6, minimal surfaces of area less than 4π exist.

The results extend classical two-dimensional geodesic theorems to higher dimensions.

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 $2 \leq n \leq 6$, 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.