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Generalized existence of extremizers for the sharp $p$-Sobolev inequality on Riemannian manifolds with nonnegative curvature

Francesco Nobili, Ivan Yuri Violo

Abstract

We study the generalized existence of extremizers for the sharp $p$-Sobolev inequality on noncompact Riemannian manifolds in connection with nonnegative curvature and Euclidean volume growth assumptions. Assuming a nonnegative Ricci curvature lower bound, we show that almost extremal functions are close in gradient norm to radial Euclidean bubbles. In the case of nonnegative sectional curvature lower bounds, we additionally deduce that vanishing is the only possible behavior, in the sense that almost extremal functions are almost zero globally. Our arguments rely on nonsmooth concentration compactness methods and Mosco-convergence results for the Cheeger energy on noncompact varying spaces, generalized to every exponent $p\in (1,\infty)$.

Generalized existence of extremizers for the sharp $p$-Sobolev inequality on Riemannian manifolds with nonnegative curvature

Abstract

We study the generalized existence of extremizers for the sharp -Sobolev inequality on noncompact Riemannian manifolds in connection with nonnegative curvature and Euclidean volume growth assumptions. Assuming a nonnegative Ricci curvature lower bound, we show that almost extremal functions are close in gradient norm to radial Euclidean bubbles. In the case of nonnegative sectional curvature lower bounds, we additionally deduce that vanishing is the only possible behavior, in the sense that almost extremal functions are almost zero globally. Our arguments rely on nonsmooth concentration compactness methods and Mosco-convergence results for the Cheeger energy on noncompact varying spaces, generalized to every exponent .

Paper Structure

This paper contains 14 sections, 21 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Calculus on ${\sf RCD}$ spaces
  4. Convergence and stability properties
  5. Alexandrov spaces: asymptotic geometry
  6. Convergence of functions on varying spaces: the case $p\neq 2$
  7. Mosco-convergence of Cheeger energies
  8. Technical results for locally Sobolev functions
  9. Concentration compactness principles
  10. Decomposition principle
  11. Proof of concentration compactness
  12. Generalized existence
  13. Nonnegative Ricci lower bound
  14. Nonnegative sectional lower bound

Key Result

Theorem 1.1

For all $\varepsilon>0, V\in(0,1), d>1$ and $p \in (1,d)$, there exists $\delta\coloneqq \delta(\varepsilon,p,d,V)>0$ such that the following holds. Let $(M,g)$ be a noncompact $d$-dimensional Riemannian manifold with ${\sf Ric}_g \ge 0$ and ${\sf AVR}(M) \in( V,1]$ and let $0\neq u\in \dot W^{1,p}( Then, there are $a\in\mathbb{R},b>0$ and $z_0 \in M$ so that

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Lemma 2.2: Density bound from reverse Sobolev inequality
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Definition 2.6: ${\sf CBB}(0)$ space
  • Theorem 2.7
  • ...and 33 more