Paper · Feb 13, 2025

3-Manifolds with positive scalar curvature and bounded geometry

Otis Chodosh, Yi Lai, Kai Xu

math.DGWider flowsarXiv:2502.09727
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Abstract

We show that a complete contractible 3-manifold with positive scalar curvature and bounded geometry must be . We also show that an open handlebody of genus larger than 1 does not admit complete metrics with positive scalar curvature and bounded geometry. Our results rely on the maximal weak solution to inverse mean curvature flow due to the third-named author.

Comments: 18 pages, 5 figures. Comments are welcome!

Mean curvature flowOther curvature flowsInverse mean curvature flow3-manifoldsScalar curvatureWeak & singular flows

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

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    Topics Inverse mean curvature flow, 3-manifolds, Scalar curvature · Same author Kai Xu · Shared terms “positive scalar”

  2. Inverse mean curvature flow with outer obstacle

    Kai XuWider flows PDF

    Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows · Same author Kai Xu

  3. Infinity-harmonic functions and inverse mean curvature flow clusters

    Kai XuWider flows PDF

    Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows · Same author Kai Xu

  4. Isoperimetry and the properness of weak inverse mean curvature flow

    Kai XuWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows · Same author Kai Xu

  5. Scalar curvature bounds for 3D continuous metrics through the Inverse Mean Curvature Flow

    Mattia Fogagnolo, Giorgio Gatti and Alessandra PludaWider flows PDF

    Topics Inverse mean curvature flow, 3-manifolds, Scalar curvature

  6. Global weak solutions for the inverse mean curvature flow in the Heisenberg group

    Adriano Pisante and Eugenio VecchiWider flows PDF

    Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows

  7. The streamlines of -harmonic functions obey the inverse mean curvature flow

    Roger MoserWider flows PDF

    Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows

  8. Classification of ruled surfaces as homothetic self-similar solutions of the inverse mean curvature flow in the Lorentz-Minkowski 3-space

    Gregório Silva Neto and Vanessa SilvaWider flows PDF

    Topics Inverse mean curvature flow, 3-manifolds, Other curvature flows

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) 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.