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15 July 2019 Minimal 2-spheres in 3-spheres
Robert Haslhofer, Daniel Ketover
Duke Math. J. 168(10): 1929-1975 (15 July 2019). DOI: 10.1215/00127094-2019-0009

Abstract

We prove that any manifold diffeomorphic to S3 and endowed with a generic metric contains at least two embedded minimal 2-spheres. The existence of at least one minimal 2-sphere was obtained by Simon and Smith in 1983. Our approach combines ideas from min–max theory and mean curvature flow. We also establish the existence of smooth mean convex foliations in 3-manifolds. We apply our methods to solve a problem posed by S. T. Yau in 1987 on whether the planar 2-spheres are the only minimal spheres in ellipsoids centered about the origin in R4. Finally, considering the example of degenerating ellipsoids, we show that the assumptions in the multiplicity 1 conjecture and the equidistribution of widths conjecture are in a certain sense sharp.

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Robert Haslhofer. Daniel Ketover. "Minimal 2-spheres in 3-spheres." Duke Math. J. 168 (10) 1929 - 1975, 15 July 2019. https://doi.org/10.1215/00127094-2019-0009

Information

Received: 11 September 2017; Revised: 7 January 2019; Published: 15 July 2019
First available in Project Euclid: 5 July 2019

zbMATH: 07108023
MathSciNet: MR3983295
Digital Object Identifier: 10.1215/00127094-2019-0009

Subjects:
Primary: 49Q05
Secondary: 49J35, 53C44, 58E12

Rights: Copyright © 2019 Duke University Press

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Vol.168 • No. 10 • 15 July 2019
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