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Quantitative C 1-stability of spheres in rank one symmetric spaces of non-compact type

  • Lauro Silini ORCID logo EMAIL logo
Published/Copyright: November 17, 2024

Abstract

We prove that in any rank one symmetric space of non-compact type M { H n , H m , H m , 𝕆 H 2 } , geodesic spheres are uniformly quantitatively stable with respect to small C 1 -volume preserving perturbations. We quantify the gain of perimeter in terms of the W 1 , 2 -norm of the perturbation, taking advantage of the explicit spectral gap of the Laplacian on geodesic spheres in M. As a consequence, we give a quantitative proof that for small volumes, geodesic spheres are isoperimetric regions among all sets of finite perimeter.


Communicated by Zoltan Balogh


Award Identifier / Grant number: 721675

Funding statement: The author has received funding from the European Research Council under the Grant Agreement No. 721675 “Regularity and Stability in Partial Differential Equations (RSPDE)”.

Acknowledgements

The author would like to thank Professors A. Figalli and U. Lang for their guidance and constant support.

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Received: 2023-05-17
Accepted: 2024-09-24
Published Online: 2024-11-17
Published in Print: 2025-01-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

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