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On mean curvature flow solitons in the sphere

Marco Magliaro, Luciano Mari, Fernanda Roing, Andreas Savas-Halilaj

TL;DR

This work analyzes solitons of the mean curvature flow in the unit sphere $\mathbb{S}^{2n+1}$ moving along the Hopf vector field. It constructs a complete, non-minimal Hopf soliton with topology $\mathbb{S}^{2n-1}\times\mathbb{R}$ that wraps around a Clifford torus $T_{2n-1,1}$ and exhibits reflection symmetry with sign-changing mean curvature, and proves a rigidity/pinching dichotomy showing that complete 2D Hopf solitons with nonnegative mean curvature outside a compact set must cover a Clifford torus. The authors develop a rotationally symmetric ansatz, reduce to a planar ODE system, establish endpoint behavior and monotonic quantities, and use these to prove key results (including Theorem THMA) about soliton structure. They further obtain rigidity results (Theorems THMC, THMB, THMD) that classify Hopf solitons under sign, curvature, and maximum-principle constraints, highlighting Clifford tori as distinguished solitons and linking geometric PDE methods with global submanifold classifications. Overall, the paper advances the classification of Hopf MCF solitons in odd-dimensional spheres and clarifies when nonminimal examples arise versus when Clifford tori appear as rigid, canonical objects.

Abstract

In this paper, we consider soliton solutions of the mean curvature flow in the unit sphere $S^{2n+1}$ moving along the integral curves of the Hopf unit vector field. While such solitons must necessarily be minimal if compact, we produce a non-minimal, complete example with topology $S^{2n-1} \times R$. The example wraps around a Clifford torus $S^{2n-1} \times S^1$ along each end, it has reflection and rotational symmetry and its mean curvature changes sign on each end. Indeed, we prove that a complete 2-dimensional soliton with non-negative mean curvature outside a compact set must be a covering of a Clifford torus. Concluding, we obtain a pinching theorem under suitable conditions on the second fundamental form.

On mean curvature flow solitons in the sphere

TL;DR

This work analyzes solitons of the mean curvature flow in the unit sphere moving along the Hopf vector field. It constructs a complete, non-minimal Hopf soliton with topology that wraps around a Clifford torus and exhibits reflection symmetry with sign-changing mean curvature, and proves a rigidity/pinching dichotomy showing that complete 2D Hopf solitons with nonnegative mean curvature outside a compact set must cover a Clifford torus. The authors develop a rotationally symmetric ansatz, reduce to a planar ODE system, establish endpoint behavior and monotonic quantities, and use these to prove key results (including Theorem THMA) about soliton structure. They further obtain rigidity results (Theorems THMC, THMB, THMD) that classify Hopf solitons under sign, curvature, and maximum-principle constraints, highlighting Clifford tori as distinguished solitons and linking geometric PDE methods with global submanifold classifications. Overall, the paper advances the classification of Hopf MCF solitons in odd-dimensional spheres and clarifies when nonminimal examples arise versus when Clifford tori appear as rigid, canonical objects.

Abstract

In this paper, we consider soliton solutions of the mean curvature flow in the unit sphere moving along the integral curves of the Hopf unit vector field. While such solitons must necessarily be minimal if compact, we produce a non-minimal, complete example with topology . The example wraps around a Clifford torus along each end, it has reflection and rotational symmetry and its mean curvature changes sign on each end. Indeed, we prove that a complete 2-dimensional soliton with non-negative mean curvature outside a compact set must be a covering of a Clifford torus. Concluding, we obtain a pinching theorem under suitable conditions on the second fundamental form.

Paper Structure

This paper contains 13 sections, 7 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Hopf solitons
  3. Hopf vector fields
  4. Hopf solitons
  5. Rotationally symmetric Hopf solitons
  6. Derivation of the equation
  7. First properties of the solutions
  8. Construction of solutions issuing from points of $\partial D \cap \{u>0\}$
  9. Further properties of solutions to \ref{['pb']}
  10. Rigidity theorems
  11. Proof of Theorem \ref{['THMC']}
  12. Proof of Theorem \ref{['THMB']}
  13. Proof of Theorem \ref{['THMD']}

Key Result

Theorem A

For each $n \ge 1$, there exists a complete, non-minimal Hopf soliton $f : M \to \mathbb{S}^{2n+1}$ which is diffeomorphic to $\mathbb{S}^{2n-1} \times \mathbb{R}$ and has the following properties:

Theorems & Definitions (10)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 4.1
  • proof