Paper · Jul 4, 2023

A topological gap theorem for the -systole of positive scalar curvature 3-manifolds

Kai Xu

math.DGWider flowsarXiv:2307.01922v3
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Abstract

Let be a closed orientable 3-manifold with scalar curvature greater than or equal to 1. If has nonvanishing second homotopy group, then it is known that the -systole of (i.e. the minimal achievable area of homotopically nontrivial spheres) is at most . We prove the following gap theorem: if is further not a quotient of , then the -systole of is no greater than an improved constant . This statement follows as a new topological application of Huisken and Ilmanen's weak inverse mean curvature flow.

Comments: v3 update: proof details added, presentation imporoved. 13 pages, 3 figures. Final version, accepted by Duke Math J

Mean curvature flowOther curvature flowsInverse mean curvature flow3-manifoldsScalar curvature

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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) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  2. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.