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On the existence of minimal Heegaard surfaces

Ketover, Daniel and Liokumovich, Yevgeny and Song, Antoine (2019) On the existence of minimal Heegaard surfaces. . (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20221026-539131000.7

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Abstract

Let H be a strongly irreducible Heegaard surface in a closed oriented Riemannian 3-manifold. We prove that H is either isotopic to a minimal surface of index at most one or isotopic to the boundary of a tubular neighborhood about a non-orientable minimal surface with a vertical handle attached. This confirms a long-standing conjecture of J. Pitts and J.H. Rubinstein. In the case of positive scalar curvature, we show for spherical space forms not diffeomorphic to S³ or RP³ that any strongly irreducible Heegaard splitting is isotopic to a minimal surface, and that there is a minimal Heegaard splitting of area less than $4π$ if R ≥ 6.


Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription
https://doi.org/10.48550/arXiv.1911.07161DOIDiscussion Paper
Additional Information:D.K. was partially supported by an NSF Postdoctoral Research fellowship as well as ERC-2011-StG-278940. Y.L. was partially supported by NSF DMS-1711053 and NSERC Discovery grants. A.S. was partially supported by NSF-DMS-1509027. This research was partially conducted during the period A.S. served as a Clay Research Fellow.
Funders:
Funding AgencyGrant Number
NSF Postdoctoral FellowshipUNSPECIFIED
European Research Council (ERC)278940
NSFDMS-1711053
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
NSFDMS-1509027
Clay Mathematics InstituteUNSPECIFIED
DOI:10.48550/arXiv.1911.07161
Record Number:CaltechAUTHORS:20221026-539131000.7
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20221026-539131000.7
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:117595
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:27 Oct 2022 21:27
Last Modified:27 Oct 2022 21:27

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