Type-II Singularities of Immersed Locally Convex Curves Evolving under Generalized Anisotropic Gage Area-Preserving Flows with Slow Curvature Speeds
Abstract
In this paper, we study the singularity behavior of immersed locally convex closed plane curves evolving under generalized anisotropic Gage area-preserving flows with slow curvature speeds of the form κσα, where 0<α≤ 1. Here κσ denotes the anisotropic curvature associated with a prescribed anisotropy function σ. We prove that every finite-time blow-up is of Type-II provided that the Euclidean length of the evolving curve remains uniformly bounded away from zero. This result holds without any symmetry assumptions on either the initial curve or the anisotropy. When the initial signed area is nonzero, this length lower bound is automatically satisfied, and thus any finite-time singularity is necessarily of Type-II. By means of Hamilton's rescaling method combined with a differential Harnack estimate, we show that along a suitable blow-up subsequence, the rescaled curves at the selected times converge locally to the profile of a translating solution of the corresponding anisotropic curvature flow V=κσα. We also establish sufficient conditions for finite-time singularity formation in terms of the initial signed area and anisotropic length. Moreover, we construct locally convex initial curves arbitrarily close to the Wulff curve for which a finite-time Type-II singularity occurs.
Related
- Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows
- Sphere Theorems for Immersed Hypersurfaces with Constant Scalar Curvature
- Normal curvature of immersed tori of dimension at most $18$
- Rigidity of Immersed Gauss Curvature Shrinkers in Euclidean Space
- The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow
BibTeX
@misc{ye2026typeii,
title = {{Type-II Singularities of Immersed Locally Convex Curves Evolving under Generalized Anisotropic Gage Area-Preserving Flows with Slow Curvature Speeds}},
author = {Ze-Yu Ye and Xiao-Liu Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35064},
eprint = {2610.35064},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35064},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Type-II Singularities of Immersed Locally Convex Curves Evolving under Generalized Anisotropic Gage Area-Preserving Flows with Slow Curvature Speeds
AU - Ze-Yu Ye
AU - Xiao-Liu Wang
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35064
UR - https://arxiv.org/abs/2610.35064
AB - In this paper, we study the singularity behavior of immersed locally convex closed plane curves evolving under generalized anisotropic Gage area-preserving flows with slow curvature speeds of the form κ_σ^α, where 0<α≤ 1. Here κ_σ denotes the anisotropic curvature associated with a prescribed anisotropy function σ. We prove that every finite-time blow-up is of Type-II provided that the Euclidean length of the evolving curve remains uniformly bounded away from zero. This result holds without any symmetry assumptions on either the initial curve or the anisotropy. When the initial signed area is nonzero, this length lower bound is automatically satisfied, and thus any finite-time singularity is necessarily of Type-II. By means of Hamilton's rescaling method combined with a differential Harnack estimate, we show that along a suitable blow-up subsequence, the rescaled curves at the selected times converge locally to the profile of a translating solution of the corresponding anisotropic curvature flow V=κ_σ^α. We also establish sufficient conditions for finite-time singularity formation in terms of the initial signed area and anisotropic length. Moreover, we construct locally convex initial curves arbitrarily close to the Wulff curve for which a finite-time Type-II singularity occurs.
ER -