eprintid: 10205710
rev_number: 8
eprint_status: archive
userid: 699
dir: disk0/10/20/57/10
datestamp: 2025-03-07 13:03:00
lastmod: 2025-03-07 13:05:12
status_changed: 2025-03-07 13:03:00
type: article
metadata_visibility: show
sword_depositor: 699
creators_name: Karpukhin, Mikhail
creators_name: Nahon, Mickaël
creators_name: Polterovich, Iosif
creators_name: Stern, Daniel
title: Stability of isoperimetric inequalities for Laplace eigenvalues on surfaces
ispublished: pub
divisions: UCL
divisions: B04
divisions: C06
divisions: F59
note: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
abstract: We prove stability estimates for the isoperimetric inequalities for the first and the second nonzero Laplace eigenvalues on surfaces, both globally and in a fixed conformal class. We employ the notion of eigenvalues of measures and show that if a normalized eigenvalue is close to its maximal value, the corresponding measure must be close in the Sobolev space W−1,2 to the set of maximizing measures. In particular, this implies a qualitative stability result: metrics almost maximizing the normalized eigenvalue must be W−1,2–close to a maximal metric. Following this approach, we prove sharp quantitative stability of the celebrated Hersch’s inequality for the first eigenvalue on the sphere, as well as of its counterpart for the second eigenvalue. Similar results are also obtained for the precise isoperimetric eigenvalue inequalities on the projective plane, torus, and Klein bottle. The square of the W−1,2 distance to a maximizing measure in these stability estimates is controlled by the difference between the normalized eigenvalue and its maximal value, indicating that the maxima are in a sense nondegenerate. We construct examples showing that the power of the distance can not be improved, and that the choice of the Sobolev space W−1,2 is optimal.
date: 2025-02
date_type: published
publisher: International Press of Boston
official_url: https://doi.org/10.4310/jdg/1738163208
oa_status: green
full_text_type: other
language: eng
primo: open
primo_central: open_green
verified: verified_manual
elements_id: 2363801
doi: 10.4310/jdg/1738163208
lyricists_name: Karpukhin, Mikhail
lyricists_id: MKARP96
actors_name: Karpukhin, Mikhail
actors_id: MKARP96
actors_role: owner
full_text_status: public
publication: Journal of Differential Geometry
volume: 129
number: 2
pagerange: 415-490
issn: 0022-040X
citation:        Karpukhin, Mikhail;    Nahon, Mickaël;    Polterovich, Iosif;    Stern, Daniel;      (2025)    Stability of isoperimetric inequalities for Laplace eigenvalues on surfaces.                   Journal of Differential Geometry , 129  (2)   pp. 415-490.    10.4310/jdg/1738163208 <https://doi.org/10.4310/jdg%2F1738163208>.       Green open access   
 
document_url: https://discovery.ucl.ac.uk/id/eprint/10205710/1/stability_submit_v2.pdf