Paper · Dec 12, 2023

Mean Curvature Flow and Heegaard Surfaces in Lens Spaces

Reto Buzano, Sylvain Maillot

math.DGWider flowsarXiv:2312.07232v2Communications in Analysis and Geometry 2026
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Abstract

We prove that the moduli space of mean convex two-spheres embedded in complete, orientable 3-dimensional Riemannian manifolds with nonnegative Ricci curvature is path-connected. This result is sharp in the sense that neither of the conditions of (strict) mean convexity, completeness, and nonnegativity of the Ricci curvature can be dropped or weakened. We also study the number of path components of mean convex Heegaard tori, again in ambient manifolds with nonnegative Ricci curvature. We prove that there are always either one or two path components and this number does not only depend on the homotopy type of the ambient manifold. We give a precise characterisation of the two cases and also discuss what happens if the mean convexity condition is weakened to nonnegative mean curvature.

Comments: v2: 21 pages, final version with minor modifications, to appear in Commun. Anal. Geom

Mean curvature flow3-manifolds

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    Topics 3-manifolds · Shared terms “nonnegative Ricci”, “completeness”, “Riemannian manifolds”

  2. Minkowski inequality in Cartan-Hadamard manifolds

    Mohammad Ghomi and Joel SpruckWider flows PDF

    Topics 3-manifolds

  3. Noncompact self-shrinkers for mean curvature flow with arbitrary genus

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    Topics Mean curvature flow · Same author Reto Buzano

  4. Isoparametric hypersurfaces of Riemannian manifolds as initial data for the mean curvature flow

    Felippe Guimarães, João Batista Marques dos Santos and João Paulo dos SantosWider flows PDF

    Topics 3-manifolds · Shared terms “Riemannian manifolds”

  5. Formation and structural stability of nondegenerate neckpinches

    Kyeongsu Choi, Panagiota Daskalopoulos and Natasa SesumWider flows PDF

    Topics 3-manifolds

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

    Kai XuWider flows PDF

    Topics 3-manifolds · Shared terms “homotopy”, “orientable”

  7. The mean curvature flow on solvmanifolds

    Romina M. Arroyo, Gabriela P. Ovando, Raquel Perales et al.Wider flows PDF

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

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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) 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.