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Embeddedness of least area minimal hypersurfaces

Song, Antoine (2018) Embeddedness of least area minimal hypersurfaces. Journal of Differential Geometry, 110 (2). pp. 345-377. ISSN 0022-040X. doi:10.4310/jdg/1538791246.

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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 ≤ 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.

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Additional Information:© 2018 Lehigh University. Received: 17 February 2016; Published: October 2018. I am grateful to my advisor Fernando Codá Marques for bringing a version of the main question to my attention. I would like to thank him for his constant support, for stimulating discussions and for guiding me through the recent literature. I also want to thank Harold Rosenberg for a meaningful discussion.
Issue or Number:2
Record Number:CaltechAUTHORS:20221026-539158000.15
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Official Citation:Antoine Song. "Embeddedness of least area minimal hypersurfaces." J. Differential Geom. 110 (2) 345 - 377, October 2018.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:117601
Deposited By: George Porter
Deposited On:27 Oct 2022 21:46
Last Modified:27 Oct 2022 21:46

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