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Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds

Francesco Nobili, Ivan Yuri Violo

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.

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.
Paper Structure (24 sections, 42 theorems, 220 equations)

This paper contains 24 sections, 42 theorems, 220 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Calculus on metric measure spaces
  4. RCD-spaces
  5. Sobolev inequalities
  6. Convergence and stability under pmGH-convergence
  7. Pólya-Szegő inequality
  8. Non-compact case
  9. Rigidity
  10. Regularity of extremal functions
  11. Rigidity of extremal functions in the Sobolev inequality
  12. Compact case
  13. Non-compact case
  14. Compactness of extremizing sequences
  15. Density upper bound
  16. ...and 9 more sections

Key Result

Theorem 1.1

For every $\varepsilon>0$ and $n >2$ there exists $\delta\coloneqq \delta(\varepsilon,n)>0$ such that the following holds. Let $(M,g)$ be an $n$-dimensional Riemannian manifold with ${\rm Ric}_g\ge (n-1)g$ and suppose there exists $u\in W^{1,2}(M)$ non-constant satisfying Then, there exist $a \in \mathbb{R}$, $b \in [0,1)$ and $z \in M$ such that Moreover, if $w_{a,b,z}\equiv a$ (i.e. $b=0$), th

Theorems & Definitions (78)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Definition 2.1
  • Lemma 2.2
  • Theorem 2.3: Coarea formula
  • proof
  • ...and 68 more