Paper · Sep 16, 2026

Eventual Monotonicity of the Willmore Energy under Mean Curvature Flow

Yuyue Chen, Shiying Wu

math.DGWider flowsarXiv:2609.18378v2
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Abstract

We establish the eventual monotonicity of the Willmore deficit along the mean curvature flow of smooth closed embedded strictly convex surfaces in , thereby complementing Huisken's exponential convergence with an eventual monotonicity property. As a consequence, the Hawking mass is eventually nondecreasing along the contracting mean curvature flow, in contrast with the usual inclusion monotonicity expected of the Hawking mass. We also discuss totally umbilical solutions in three-dimensional space forms, where the monotonicity directions of the Willmore energy and the Hawking mass depend on the sign of the ambient sectional curvature.

Mean curvature flow3-manifoldsSphere theoremsLong-time existence & convergenceGeometric inequalitiesPhysics

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  1. Hawking Mass Monotonicity for Initial Data Sets

    Sven HirschWider flows PDF

    Topics Physics · Shared terms “Hawking mass”

  2. The quermassintegral preserving mean curvature flow in the sphere

    Esther Cabezas-Rivas and Julian ScheuerWider flows PDF

    Topics Sphere theorems, Geometric inequalities, Physics · Shared terms “space forms”, “strictly convex”, “Huisken”

  3. Curvature quotient flows and Alexandrov–Fenchel inequalities in hyperbolic space

    Tianci Luo, Yong Wei, Ruijia Zhang et al.Wider flows PDF

    Topics Sphere theorems, Geometric inequalities, Long-time existence & convergence · Shared terms “sectional curvature”, “smooth closed”, “strictly convex”

  4. Geometric -stability of geodesic spheres under volume-preserving mean curvature flow

    Esther Cabezas-Rivas and Carlo SinestrariWider flows PDF

    Topics Sphere theorems, Long-time existence & convergence · Shared terms “eventual”, “exponential convergence”, “space forms”

  5. On the Isoperimetric Riemannian Penrose Inequality

    Luca Benatti, Mattia Fogagnolo and Lorenzo MazzieriWider flows PDF

    Topics Geometric inequalities, Physics · Shared terms “Hawking mass”, “Huisken”

  6. Minkowski inequality in Cartan-Hadamard manifolds

    Mohammad Ghomi and Joel SpruckWider flows PDF

    Topics Geometric inequalities, 3-manifolds · Shared terms “convex surfaces”

  7. The Penrose inequality with charge for 2-convex initial data sets

    Tuan DolmenWider flows PDF

    Topics Geometric inequalities, Physics · Shared terms “Hawking mass”

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

    Mattia Fogagnolo, Giorgio Gatti and Alessandra PludaWider flows PDF

    Topics Physics, 3-manifolds · Shared terms “Hawking mass”

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) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  2. 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.

  3. 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.