Welcome to the new version of CaltechAUTHORS. Login is currently restricted to library staff. If you notice any issues, please email coda@library.caltech.edu
Published October 2018 | public
Journal Article

Embeddedness of least area minimal hypersurfaces

Song, Antoine

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.

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.

Additional details

Created:
August 22, 2023
Modified:
October 24, 2023