Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds
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
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Advances in MathematicsDate
2024Copyright
© 2024 The Author(s). Published by Elsevier Inc.
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.
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https://converis.jyu.fi/converis/portal/detail/Publication/207000065
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Research Council of FinlandFunding program(s)
Academy Project, AoF; Research costs of Academy Research Fellow, AoFAdditional 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.License
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