---
OA_place: repository
OA_type: green
_id: '22340'
abstract:
- lang: eng
text: We show that Laplace isospectral deformations within a conformal class of
generic Liouville metrics on the two-dimensional torus that are linear in the
deformation parameter are necessarily trivial. Two of the main ingredients in
our proof are a noncancellation result for the wave trace and an analysis of the
second order variational formula for the energy functional associated to closed
geodesics. Noncancellation allows us to detect parts of the length spectrum from
the Laplace spectrum and conclude rational integrability for the deformed geodesic
flow (Liouville metrics are folklorically conjectured to be the only Riemannian
metrics with integrable geodesic flow on the torus). We then use the second variational
formula to show how the preservation of a single rational torus is sufficient
to conclude triviality of the deformation, assuming linearity. We also present
some evidence that our hypothesis of linearity may indeed be necessary.
article_processing_charge: No
arxiv: 1
author:
- first_name: Sven Joscha
full_name: Henheik, Sven Joscha
id: 31d731d7-d235-11ea-ad11-b50331c8d7fb
last_name: Henheik
orcid: 0000-0003-1106-327X
- first_name: Vadim
full_name: Kaloshin, Vadim
id: FE553552-CDE8-11E9-B324-C0EBE5697425
last_name: Kaloshin
orcid: 0000-0002-6051-2628
- first_name: Yunzhe
full_name: Li, Yunzhe
id: 41cb05d3-f128-11eb-9611-e4e2b3cfba31
last_name: Li
- first_name: Amir
full_name: Vig, Amir
id: 49d58dd5-45f5-11ec-9f86-8ce1276989b9
last_name: Vig
citation:
ama: Henheik SJ, Kaloshin V, Li Y, Vig A. Spectral rigidity of Liouville tori. arXiv.
doi:10.48550/ARXIV.2511.10398
apa: Henheik, S. J., Kaloshin, V., Li, Y., & Vig, A. (n.d.). Spectral rigidity
of Liouville tori. arXiv. https://doi.org/10.48550/ARXIV.2511.10398
chicago: Henheik, Sven Joscha, Vadim Kaloshin, Yunzhe Li, and Amir Vig. “Spectral
Rigidity of Liouville Tori.” ArXiv, n.d. https://doi.org/10.48550/ARXIV.2511.10398.
ieee: S. J. Henheik, V. Kaloshin, Y. Li, and A. Vig, “Spectral rigidity of Liouville
tori,” arXiv. .
ista: Henheik SJ, Kaloshin V, Li Y, Vig A. Spectral rigidity of Liouville tori.
arXiv, 10.48550/ARXIV.2511.10398.
mla: Henheik, Sven Joscha, et al. “Spectral Rigidity of Liouville Tori.” ArXiv,
doi:10.48550/ARXIV.2511.10398.
short: S.J. Henheik, V. Kaloshin, Y. Li, A. Vig, ArXiv (n.d.).
corr_author: '1'
date_created: 2026-07-14T12:51:50Z
date_published: 2025-11-13T00:00:00Z
date_updated: 2026-07-20T14:58:23Z
day: '13'
department:
- _id: VaKa
- _id: LaEr
doi: 10.48550/ARXIV.2511.10398
external_id:
arxiv:
- '2511.10398'
keyword:
- Differential Geometry (math.DG)
- Mathematical Physics (math-ph)
- Dynamical Systems (math.DS)
- Spectral Theory (math.SP)
- 'FOS: Mathematics'
- 'FOS: Mathematics'
- 'FOS: Physical sciences'
- 'FOS: Physical sciences'
- 58J42
- 37J35
- 37J35
- 35P20
- 58J40
- 58J50
- 37D40
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2511.10398
month: '11'
oa: 1
oa_version: Preprint
publication: arXiv
publication_status: draft
related_material:
record:
- id: '22255'
relation: dissertation_contains
status: public
status: public
title: Spectral rigidity of Liouville tori
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: preprint
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
year: '2025'
...
---
OA_type: closed access
_id: '3994'
abstract:
- lang: eng
text: The body defined by a finite collection of disks is a subset of the plane
bounded by a tangent continuous curve, which we call the skin. We give analytic
formulas for the area, the perimeter, the area derivative, and the perimeter derivative
of the body. Given the filtrations of the Delaunay triangulation and the Voronoi
diagram of the disks, all formulas can be evaluated in time proportional to the
number of disks.
article_processing_charge: No
article_type: original
author:
- first_name: Ho
full_name: Cheng, Ho
last_name: Cheng
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
citation:
ama: Cheng H, Edelsbrunner H. Area, perimeter and derivatives of a skin curve. Computational
Geometry. 2003;26(2):173-192. doi:10.1016/S0925-7721(02)00124-4
apa: Cheng, H., & Edelsbrunner, H. (2003). Area, perimeter and derivatives of
a skin curve. Computational Geometry. Elsevier. https://doi.org/10.1016/S0925-7721(02)00124-4
chicago: Cheng, Ho, and Herbert Edelsbrunner. “Area, Perimeter and Derivatives of
a Skin Curve.” Computational Geometry. Elsevier, 2003. https://doi.org/10.1016/S0925-7721(02)00124-4.
ieee: H. Cheng and H. Edelsbrunner, “Area, perimeter and derivatives of a skin curve,”
Computational Geometry, vol. 26, no. 2. Elsevier, pp. 173–192, 2003.
ista: Cheng H, Edelsbrunner H. 2003. Area, perimeter and derivatives of a skin curve.
Computational Geometry. 26(2), 173–192.
mla: Cheng, Ho, and Herbert Edelsbrunner. “Area, Perimeter and Derivatives of a
Skin Curve.” Computational Geometry, vol. 26, no. 2, Elsevier, 2003, pp.
173–92, doi:10.1016/S0925-7721(02)00124-4.
short: H. Cheng, H. Edelsbrunner, Computational Geometry 26 (2003) 173–192.
date_created: 2018-12-11T12:06:20Z
date_published: 2003-10-01T00:00:00Z
date_updated: 2026-05-06T08:12:09Z
day: '01'
doi: 10.1016/S0925-7721(02)00124-4
extern: '1'
intvolume: ' 26'
issue: '2'
keyword:
- Computational geometry
- Differential geometry
- Skin curves
- Voronoi diagrams
- Delaunay triangulations
- Filtrations
- Disks
- Hyperbolas
- Area
- Perimeter
- Derivatives
language:
- iso: eng
month: '10'
oa_version: None
page: 173 - 192
publication: Computational Geometry
publication_status: published
publisher: Elsevier
publist_id: '2135'
quality_controlled: '1'
status: public
title: Area, perimeter and derivatives of a skin curve
type: journal_article
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
volume: 26
year: '2003'
...