Singular Rotational Self-Similar Tori for Odd σk-Curvature Flows
Abstract
For every pair of integers 3≤ k<n with k odd, we construct a compact embedded rotational torus in ℝn+1 whose homothetic dilations satisfy the unnormalised σk-curvature flow in a Sobolev almost-everywhere sense. Its profile curve has Hölder regularity C1,1/k and Sobolev regularity W2,p for every 1≤ p<k/(k−1). Away from two singular latitudes the torus is smooth; globally, the flow equation is interpreted using the weak shape operator of the associated Lipschitz boundary. Under rotational symmetry, the self-similar equation ⟨ X,ν⟩=−σk, where X is the position vector and ν is the unit normal, reduces to a degenerate profile system. We solve this system by combining an odd-power desingularisation, a shooting argument, uniform radial and axial bounds, and a strict gap between the shooting parameters and the cylindrical radius. No classical C2 rotational torus can satisfy the soliton equation, so the loss of regularity is unavoidable within the rotational toroidal class.
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BibTeX
@misc{cheng2026singular,
title = {{Singular Rotational Self-Similar Tori for Odd $σ_k$-Curvature Flows}},
author = {Haoxuan Cheng and Junqi Lai and Guoxin Wei},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01346},
eprint = {2609.01346},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01346},
}
RIS
TY - UNPB
TI - Singular Rotational Self-Similar Tori for Odd σ_k-Curvature Flows
AU - Haoxuan Cheng
AU - Junqi Lai
AU - Guoxin Wei
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01346
UR - https://arxiv.org/abs/2609.01346
AB - For every pair of integers 3≤ k<n with k odd, we construct a compact embedded rotational torus in ℝ^n+1 whose homothetic dilations satisfy the unnormalised σ_k-curvature flow in a Sobolev almost-everywhere sense. Its profile curve has Hölder regularity C^1,1/k and Sobolev regularity W^2,p for every 1≤ p<k/(k−1). Away from two singular latitudes the torus is smooth; globally, the flow equation is interpreted using the weak shape operator of the associated Lipschitz boundary. Under rotational symmetry, the self-similar equation ⟨ X,ν⟩=−σ_k, where X is the position vector and ν is the unit normal, reduces to a degenerate profile system. We solve this system by combining an odd-power desingularisation, a shooting argument, uniform radial and axial bounds, and a strict gap between the shooting parameters and the cylindrical radius. No classical C^2 rotational torus can satisfy the soliton equation, so the loss of regularity is unavoidable within the rotational toroidal class.
ER -