TY  - JOUR
VL  - 129
SP  - 415
IS  - 2
N1  - This version is the author accepted manuscript. For information on re-use, please refer to the publisher?s terms and conditions.
UR  - https://doi.org/10.4310/jdg/1738163208
SN  - 0022-040X
A1  - Karpukhin, Mikhail
A1  - Nahon, Mickaël
A1  - Polterovich, Iosif
A1  - Stern, Daniel
JF  - Journal of Differential Geometry
AV  - public
Y1  - 2025/02//
EP  - 490
TI  - Stability of isoperimetric inequalities for Laplace eigenvalues on surfaces
PB  - International Press of Boston
N2  - 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.
ID  - discovery10205710
ER  -