Paper · Nov 9, 2020

Multiplicity one for min-max theory in compact manifolds with boundary and its applications

Ao Sun, Zhichao Wang, Xin Zhou

math.DGWider flowsarXiv:2011.04136
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As on arXiv

Abstract

We prove the multiplicity one theorem for min-max free boundary minimal hypersurfaces in compact manifolds with boundary of dimension between 3 and 7 for generic metrics. To approach this, we develop existence and regularity theory for free boundary hypersurface with prescribed mean curvature, which includes the regularity theory for minimizers, compactness theory, and a generic min-max theory with Morse index bounds. As applications, we construct new free boundary minimal hypersurfaces in the unit balls in Euclidean spaces and self-shrinkers of the mean curvature flows with arbitrarily large entropy.

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Submitted
Nov 9, 2020
Subjects
math.DG, math.AP

Comments: 50 pages. Comments are welcome!

Mean curvature flowSelf-shrinkers & translatorsLimit spaces & RCDBoundaries & non-compactFree boundary & capillaryMinimal & CMC surfaces

Its neighbourhood

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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 Mean curvature flow with free boundary in embedded cylinders or cones and uniqueness results for minimal hypersurfaces (Valentina-Mira Wheeler, 2016)7Wheeler 2016 Index and first Betti number of f-minimal hypersurfaces and self-shrinkers (Debora Impera, Michele Rimoldi and Alessandro Savo, 2018)5Impera 2018 Free boundary minimal disks in convex balls (Robert Haslhofer and Daniel Ketover, 2023)1Haslhofer 2023 The Smale conjecture and min-max theory (Daniel Ketover and Yevgeny Liokumovich, 2023)4Ketover 2023 Mean curvature flow with multiplicity 2 convergence in closed manifolds (Jingwen Chen and Ao Sun, 2024)2Chen 2024 Existence of 5 minimal tori in 3-spheres of positive Ricci curvature (Adrian Chun-Pong Chu and Yangyang Li, 2024)8Chu 2024 Existence of Constant Mean Curvature Surfaces in Asymptotically Flat and Asymptotically Hyperbolic Manifolds (Liam Mazurowski and Jintian Zhu, 2025)6Mazurowski 2025 Ancient solutions to free boundary mean curvature flow (Theodora Bourni and Giada Franz, 2026)3Bourni 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Free boundary minimal disks in convex balls

    Robert Haslhofer and Daniel KetoverWider flows

    Topics Minimal & CMC surfaces, Free boundary & capillary · Shared terms “max theory”, “min max”, “generic metrics”

  2. Mean curvature flow with multiplicity convergence in closed manifolds

    Jingwen Chen and Ao SunWider flows

    Topics Minimal & CMC surfaces · Same author Ao Sun · Shared terms “multiplicity”, “minimal hypersurfaces”

  3. Ancient solutions to free boundary mean curvature flow

    Theodora Bourni and Giada FranzWider flows

    Topics Minimal & CMC surfaces, Free boundary & capillary · Shared terms “minimal hypersurfaces”, “Morse index”

  4. The Smale conjecture and min-max theory

    Daniel Ketover and Yevgeny Liokumovich

    Topics Minimal & CMC surfaces · Shared terms “max theory”, “min max”

  5. Index and first Betti number of -minimal hypersurfaces and self-shrinkers

    Debora Impera, Michele Rimoldi and Alessandro SavoWider flows

    Topics Minimal & CMC surfaces, Self-shrinkers & translators, Boundaries & non-compact · Shared terms “minimal hypersurfaces”, “Morse index”, “Euclidean space”

  6. Existence of Constant Mean Curvature Surfaces in Asymptotically Flat and Asymptotically Hyperbolic Manifolds

    Liam Mazurowski and Jintian ZhuWider flows

    Topics Minimal & CMC surfaces, Free boundary & capillary, Boundaries & non-compact · Shared terms “max theory”, “min max”

  7. Mean curvature flow with free boundary in embedded cylinders or cones and uniqueness results for minimal hypersurfaces

    Valentina-Mira WheelerWider flows

    Topics Minimal & CMC surfaces, Free boundary & capillary · Shared terms “minimal hypersurfaces”

  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 · Shared terms “max theory”, “min max”

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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.