Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds
Abstract
We study the qualitative stability of two classes of Sobolev inequalities on Riemannian manifolds. In the case of positive Ricci curvature, we prove that an almost extremal function for the sharp Sobolev inequality is close to an extremal function of the round sphere. In the setting of non-negative Ricci curvature and Euclidean volume growth, we show an analogous result in comparison with the extremal functions in the Euclidean Sobolev inequality. As an application, we deduce a stability result for minimizing Yamabe metrics. The arguments rely on a generalized Lions' concentration compactness on varying spaces and on rigidity results of Sobolev inequalities on singular spaces.
Main Authors
Format
Articles
Research article
Published
2024
Series
Subjects
Publication in research information system
Publisher
Elsevier
The permanent address of the publication
https://urn.fi/URN:NBN:fi:jyu-202402141866Use this for linking
Review status
Peer reviewed
ISSN
0001-8708
DOI
https://doi.org/10.1016/j.aim.2024.109521
Language
English
Published in
Advances in Mathematics
Citation
- Nobili, F., & Violo, I. Y. (2024). Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds. Advances in Mathematics, 440, Article 109521. https://doi.org/10.1016/j.aim.2024.109521
License
Funder(s)
Research Council of Finland
Research Council of Finland
Funding program(s)
Academy Project, AoF
Research costs of Academy Research Fellow, AoF
Akatemiahanke, SA
Akatemiatutkijan tutkimuskulut, SA
Additional information about funding
F.N. is supported by the Academy of Finland Grant No. 314789. I.Y.V. is supported by the Academy of Finland projects Incidences on Fractals, Grant No. 321896 and Singular integrals, harmonic functions, and boundary regularity in Heisenberg groups, Grant No. 328846.
Copyright© 2024 The Author(s). Published by Elsevier Inc.