Paper · Oct 5, 2020

Rigidity of riemannian manifolds containing an equator

Laurent Mazet

math.DGWider flowsarXiv:2010.01994v3
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Abstract

In this paper, we prove that a Riemannian -manifold with sectional curvature bounded above by that contains a minimal -sphere of area which has index at least has constant sectional curvature . The proof uses the construction of ancient mean curvature flows that flow out of a minimal submanifold. As a consequence we also prove a rigidity result for the Simon-Smith minimal spheres.

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Submitted
Oct 5, 2020
Last revised
Mar 23, 2021 (v3)
Subjects
math.DG
Journal
Communications in Analysis and Geometry, Vol. 32, Issue 1 (2024), pp. 21-63
DOI
10.4310/CAG.240905212306

Comments: 33 pages, 1 figure. This new version contains a major improvement: the main novelty is a rigidity result for the min-max minimal spheres constructed by Simon and Smith

Mean curvature flowAncient solutionsStability & uniquenessMinimal & CMC surfaces

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Minimal two-spheres in three-spheres (Robert Haslhofer and Daniel Ketover, 2017)4Haslhofer 2017 Ancient and Eternal Solutions to Mean Curvature Flow from Minimal Surfaces (Alexander Mramor and Alec Payne, 2019)6Mramor 2019 Ancient solutions to mean curvature flow for isoparametric submanifolds (Xiaobo Liu and Chuu-Lian Terng, 2019)2Liu 2019 Ancient mean curvature flows from minimal hypersurfaces (Yongheng Han, 2023)1Han 2023 Ancient mean curvature flows with finite total curvature (Kyeongsu Choi, Jiuzhou Huang and Taehun Lee, 2024)5Choi 2024 Existence of 5 minimal tori in 3-spheres of positive Ricci curvature (Adrian Chun-Pong Chu and Yangyang Li, 2024)8Chu 2024 Ancient solutions to free boundary mean curvature flow (Theodora Bourni and Giada Franz, 2026)7Bourni 2026 Ancient mean curvature flow asymptotic to a minimal quadratic cone (Junyoung Park, 2026)3Park 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ancient mean curvature flows from minimal hypersurfaces

    Yongheng HanWider flows

    Topics Minimal & CMC surfaces, Ancient solutions, Stability & uniqueness

  2. Ancient solutions to mean curvature flow for isoparametric submanifolds

    Xiaobo Liu and Chuu-Lian TerngWider flows

    Topics Minimal & CMC surfaces, Ancient solutions, Stability & uniqueness

  3. Ancient mean curvature flow asymptotic to a minimal quadratic cone

    Junyoung ParkWider flows

    Topics Minimal & CMC surfaces, Ancient solutions, Stability & uniqueness · Shared terms “Simon”

  4. Minimal two-spheres in three-spheres

    Robert Haslhofer and Daniel KetoverWider flows

    Topics Minimal & CMC surfaces · Shared terms “Simon Smith”, “minimal spheres”

  5. Ancient mean curvature flows with finite total curvature

    Kyeongsu Choi, Jiuzhou Huang and Taehun LeeWider flows

    Topics Minimal & CMC surfaces, Ancient solutions

  6. Ancient and Eternal Solutions to Mean Curvature Flow from Minimal Surfaces

    Alexander Mramor and Alec PayneWider flows

    Topics Minimal & CMC surfaces, Ancient solutions, Stability & uniqueness

  7. Ancient solutions to free boundary mean curvature flow

    Theodora Bourni and Giada FranzWider flows

    Topics Minimal & CMC surfaces, Ancient solutions, Stability & uniqueness

  8. Existence of 5 minimal tori in 3-spheres of positive Ricci curvature

    Adrian Chun-Pong Chu and Yangyang LiWider flows

    Topics Minimal & CMC surfaces

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979)

In the guidebook

  1. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  2. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

    Ancient, nonnegatively curved, noncollapsed: the models for singularities.

  3. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.