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October 2018 Embeddedness of least area minimal hypersurfaces
Antoine Song
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J. Differential Geom. 110(2): 345-377 (October 2018). DOI: 10.4310/jdg/1538791246

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 $2 \leq n \leq 6$, 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.

Citation

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Antoine Song. "Embeddedness of least area minimal hypersurfaces." J. Differential Geom. 110 (2) 345 - 377, October 2018. https://doi.org/10.4310/jdg/1538791246

Information

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

zbMATH: 06958643
MathSciNet: MR3861813
Digital Object Identifier: 10.4310/jdg/1538791246

Rights: Copyright © 2018 Lehigh University

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Vol.110 • No. 2 • October 2018
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