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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 (Unpublished)

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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)
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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.
Funding AgencyGrant Number
NSF Postdoctoral FellowshipUNSPECIFIED
European Research Council (ERC)278940
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Clay Mathematics InstituteUNSPECIFIED
Record Number:CaltechAUTHORS:20221026-539131000.7
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:117595
Deposited By: George Porter
Deposited On:27 Oct 2022 21:27
Last Modified:27 Oct 2022 21:27

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