Paper · Sep 14, 2024

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

Adrian Chun-Pong Chu, Yangyang Li

math.DGWider flowsarXiv:2409.09315v2
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Abstract

In 1989, B. White conjectured that every Riemannian 3-sphere has at least 5 embedded minimal tori. We confirm this conjecture for 3-spheres of positive Ricci curvature. While our proof uses min-max theory, the underlying heuristics are largely inspired by mean curvature flow.

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Submitted
Sep 14, 2024
Last revised
Jul 31, 2025 (v2)
Subjects
math.DG, math.AP, math.GT

Comments: v2: Minor revisions, revised section 8.1

Mean curvature flowMinimal & CMC surfaces

Its neighbourhood

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
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    Topics Minimal & CMC surfaces · Shared terms “least embedded”, “max theory”, “min max”

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    Topics Minimal & CMC surfaces

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

  4. On the long-time limit of the mean curvature flow in closed manifolds

    Alexander Mramor and Ao SunWider flows

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

  5. Mean curvature flow with multiplicity convergence in closed manifolds

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    Topics Minimal & CMC surfaces

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    Katherine Castro and César RosalesWider flows

    Topics Minimal & CMC surfaces

  7. Ancient mean curvature flows from minimal hypersurfaces

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    Topics Minimal & CMC surfaces

  8. The Dirichlet problem for the minimal surface system on smooth domains

    Caiyan Li and Hengyu ZhouWider 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. 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.