Paper · Jul 4, 2025

When can minimal hypersurfaces be connected by mean curvature flow?

Jingwen Chen, Pedro Gaspar

math.DGWider flowsarXiv:2507.03837
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Abstract

From the perspective of Morse theory, it is natural to investigate gradient flow trajectories between critical points. In this short note, we explore the minimal hypersurface analogue of this phenomenon and present examples that suggest additional topological and variational obstructions to the existence of connecting mean curvature flows.

Comments: 7 pages. Comments are welcome

Mean curvature flowMinimal & 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 2021, 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 and low energy solutions of the parabolic Allen-Cahn equation on the three-sphere (Jingwen Chen and Pedro Gaspar, 2021)6Chen 2021 Uniqueness results and enclosure properties for hypersurfaces with boundary in weighted cylinders (Katherine Castro and César Rosales, 2022)5Castro 2022 Ancient mean curvature flows from minimal hypersurfaces (Yongheng Han, 2023)2Han 2023 Mean curvature flow with multiplicity 2 convergence in closed manifolds (Jingwen Chen and Ao Sun, 2024)1Chen 2024 Existence of 5 minimal tori in 3-spheres of positive Ricci curvature (Adrian Chun-Pong Chu and Yangyang Li, 2024)4Chu 2024 Remarks on minimal hypersurfaces in shrinking gradient Ricci solitons (Yukai Sun and Guangrui Zhu, 2025)3Sun 2025 Ancient solutions to free boundary mean curvature flow (Theodora Bourni and Giada Franz, 2026)8Bourni 2026 Distance between minimal surfaces and flows (Tobias Holck Colding and William P. Minicozzi, 2026)7Colding 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Mean curvature flow with multiplicity convergence in closed manifolds

    Jingwen Chen and Ao SunWider flows PDF

    Topics Minimal & CMC surfaces · Same author Jingwen Chen · Shared terms “minimal hypersurface”

  2. Ancient mean curvature flows from minimal hypersurfaces

    Yongheng HanWider flows PDF

    Topics Minimal & CMC surfaces · Shared terms “minimal hypersurface”, “Morse”

  3. Remarks on minimal hypersurfaces in shrinking gradient Ricci solitons

    Yukai Sun and Guangrui Zhu PDF

    Topics Minimal & CMC surfaces · Shared terms “minimal hypersurface”

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

    Adrian Chun-Pong Chu and Yangyang LiWider flows PDF

    Topics Minimal & CMC surfaces

  5. Uniqueness results and enclosure properties for hypersurfaces with boundary in weighted cylinders

    Katherine Castro and César RosalesWider flows PDF

    Topics Minimal & CMC surfaces · Shared terms “minimal hypersurface”

  6. Mean curvature flow and low energy solutions of the parabolic Allen-Cahn equation on the three-sphere

    Jingwen Chen and Pedro GasparWider flows PDF

    Topics Mean curvature flow · Same authors Jingwen Chen, Pedro Gaspar · Shared terms “gradient flow”, “Morse”

  7. Distance between minimal surfaces and flows

    Tobias Holck Colding and William P. MinicozziWider flows PDF

    Topics Minimal & CMC surfaces · Shared terms “minimal hypersurface”

  8. Ancient solutions to free boundary mean curvature flow

    Theodora Bourni and Giada FranzWider flows PDF

    Topics Minimal & CMC surfaces · Shared terms “minimal hypersurface”, “Morse”

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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.