Embeddedness of least area minimal hypersurfaces

Abstract

In “Simple closed geodesics on convex surfaces” [J. Differential Geom., 36(3):517–549, 1992], 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 ≤ n ≤ 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 4π. This confirms a conjecture of F. C. Marques and A. Neves.