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PreprintCC BY 4.0arXiv

Solitons for the inverse curve shortening flow on the hyperbolic plane

  • Ivan Krznarić
  • Rafael López
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Abstract

We study the solitons of the inverse curve shortening flow in the hyperbolic plane with respect to Killing fields: all parabolic fields, and the hyperbolic and elliptic fields with unit translation length and unit angular speed. The three cases reduce to one planar system. Except for a family of horocycles, every soliton has an endpoint at finite distance where the curvature blows up and the curve is a branch of a cusp. In the parabolic and elliptic cases there is a critical value H of the parameter of the family: below H the other end meets the ideal boundary orthogonally, above H it is a second singular endpoint, and at H the curvature tends to $1$. In the hyperbolic case H=π/2 and the critical solitons are horocycles. These horocycles are the only complete solitons and the only solitons of constant curvature.

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Cite

BibTeXRIS
BibTeX
@misc{krznaric2026solitons,
  title = {{Solitons for the inverse curve shortening flow on the hyperbolic plane}},
  author = {Ivan Krznarić and Rafael López},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2605.14385},
  eprint = {2605.14385},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.14385},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Solitons for the inverse curve shortening flow on the hyperbolic plane
AU  - Ivan Krznarić
AU  - Rafael López
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2605.14385
UR  - https://arxiv.org/abs/2605.14385
AB  - We study the solitons of the inverse curve shortening flow in the hyperbolic plane with respect to Killing fields: all parabolic fields, and the hyperbolic and elliptic fields with unit translation length and unit angular speed. The three cases reduce to one planar system. Except for a family of horocycles, every soliton has an endpoint at finite distance where the curvature blows up and the curve is a branch of a cusp. In the parabolic and elliptic cases there is a critical value H of the parameter of the family: below H the other end meets the ideal boundary orthogonally, above H it is a second singular endpoint, and at H the curvature tends to $1$. In the hyperbolic case H=π/2 and the critical solitons are horocycles. These horocycles are the only complete solitons and the only solitons of constant curvature.
ER  -