Area-preserving curvature flow for open curves with general contact angles on skew lines
Abstract
We study an area-preserving curvature flow for planar open curves whose end points lie on skew lines and meet the lines at general contact angles. In previous studies of moving boundary problems for higher-order geometric gradient flows, the right-angle condition has often been imposed to ensure that the evolving curves naturally satisfy the boundary conditions. However, under general contact-angle conditions, the evolution must be described by a free boundary problem. Under suitable assumptions on the contact angles θ±, the angle θ between the skew lines, and the initial data γ0, we prove global-in-time existence and uniqueness up to reparametrization. We further prove that, after choosing a constant-speed parametrization at each time, the solution converges exponentially to a circular arc with the same signed area as the initial curve. Our analysis is based on a formulation of the evolution as a free boundary problem together with the derivation of uniform a priori estimates.
Related
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BibTeX
@misc{hiroi2026areapreserving,
title = {{Area-preserving curvature flow for open curves with general contact angles on skew lines}},
author = {Fuya Hiroi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05909},
eprint = {2610.05909},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05909},
}
RIS
TY - UNPB
TI - Area-preserving curvature flow for open curves with general contact angles on skew lines
AU - Fuya Hiroi
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.05909
UR - https://arxiv.org/abs/2610.05909
AB - We study an area-preserving curvature flow for planar open curves whose end points lie on skew lines and meet the lines at general contact angles. In previous studies of moving boundary problems for higher-order geometric gradient flows, the right-angle condition has often been imposed to ensure that the evolving curves naturally satisfy the boundary conditions. However, under general contact-angle conditions, the evolution must be described by a free boundary problem. Under suitable assumptions on the contact angles θ_±, the angle θ between the skew lines, and the initial data γ_0, we prove global-in-time existence and uniqueness up to reparametrization. We further prove that, after choosing a constant-speed parametrization at each time, the solution converges exponentially to a circular arc with the same signed area as the initial curve. Our analysis is based on a formulation of the evolution as a free boundary problem together with the derivation of uniform a priori estimates.
ER -