Journal of Differential Geometry

Embeddedness of least area minimal hypersurfaces

Antoine Song

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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 $2 \leq n \leq 6$, 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.

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J. Differential Geom., Volume 110, Number 2 (2018), 345-377.

Received: 17 February 2016
First available in Project Euclid: 6 October 2018

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Song, Antoine. Embeddedness of least area minimal hypersurfaces. J. Differential Geom. 110 (2018), no. 2, 345--377. doi:10.4310/jdg/1538791246.

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