Comparison Theorems for the Profile Curve Equation of Rotationally Symmetric Self-Shrinkers

Abstract

Mean curvature flow is a fundamental geometric evolution equation in which a submanifold moves in the normal direction with velocity equal to its mean curvature vector. Self-shrinkers arise naturally as self-similar solutions to the mean curvature flow and play an important role as models for finite-time singularities. Among nontrivial examples of compact embedded self-shrinkers, the rotationally symmetric self-shrinking torus constructed by Angenent is one of the most important. However, the uniqueness of the Angenent torus remains a major open problem. In this paper, we study rotationally symmetric self-shrinkers of type 𝕊¹× 𝕊ⁿ⁻¹ from the point of view of ordinary differential equations. We analyze the profile curves of rotationally symmetric self-shrinkers, focusing on the behavior of their vertical points and the curves traced out by these points as the initial height varies. We give a new proof of the existence of the Angenent torus by showing that two families of vertical-point trajectories must intersect. We further derive the linearized equation associated with the rotationally symmetric self-shrinker equation and apply a Sturm-type comparison theorem to obtain sufficient conditions for the monotonicity of horizontal-point trajectories. In particular, we prove a comparison theorem for solutions near the spherical self-shrinker x²+r²=2n, and establish partial monotonicity results for the curves of horizontal points. These results provide a possible approach to the uniqueness problem for the Angenent torus.

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BibTeX RIS

BibTeX

@misc{peng2026comparison,
  title = {{Comparison Theorems for the Profile Curve Equation of Rotationally Symmetric Self-Shrinkers}},
  author = {Peng Peng},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.20276},
  eprint = {2606.20276},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.20276},
}

RIS

TY  - UNPB
TI  - Comparison Theorems for the Profile Curve Equation of Rotationally Symmetric Self-Shrinkers
AU  - Peng Peng
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2606.20276
UR  - https://arxiv.org/abs/2606.20276
AB  - Mean curvature flow is a fundamental geometric evolution equation in which a submanifold moves in the normal direction with velocity equal to its mean curvature vector. Self-shrinkers arise naturally as self-similar solutions to the mean curvature flow and play an important role as models for finite-time singularities. Among nontrivial examples of compact embedded self-shrinkers, the rotationally symmetric self-shrinking torus constructed by Angenent is one of the most important. However, the uniqueness of the Angenent torus remains a major open problem. In this paper, we study rotationally symmetric self-shrinkers of type 𝕊^1× 𝕊^n−1 from the point of view of ordinary differential equations. We analyze the profile curves of rotationally symmetric self-shrinkers, focusing on the behavior of their vertical points and the curves traced out by these points as the initial height varies. We give a new proof of the existence of the Angenent torus by showing that two families of vertical-point trajectories must intersect. We further derive the linearized equation associated with the rotationally symmetric self-shrinker equation and apply a Sturm-type comparison theorem to obtain sufficient conditions for the monotonicity of horizontal-point trajectories. In particular, we prove a comparison theorem for solutions near the spherical self-shrinker x^2+r^2=2n, and establish partial monotonicity results for the curves of horizontal points. These results provide a possible approach to the uniqueness problem for the Angenent torus.
ER  -

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