Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fractional Yamabe problem on locally flat conformal infinities of Poincare-Einstein manifolds

Martin Mayer, Cheikh Birahim Ndiaye

Abstract

We study in this paper the fractional Yamabe problem first considered by Gonzalez-Qing on the conformal infinity $(M^n , [h])$ of a Poincaré-Einstein manifold $(X^{n+1} , g^+ )$ with either $n = 2$ or $n \geq 3$ and $(M^n , [h])$ is locally flat - namely $(M, h)$ is locally conformally flat. However, as for the classical Yamabe problem, because of the involved quantization phenomena, the variational analysis of the fractional one exhibits also a local situation and a global one. Furthermore the latter global situation includes the case of conformal infinities of Poincaré-Einstein manifolds of dimension either 2 or of dimension greater than $2$ and which are locally flat, and hence the minimizing technique of Aubin- Schoen in that case clearly requires an analogue of the positive mass theorem of Schoen-Yau which is not known to hold. Using the algebraic topological argument of Bahri-Coron, we bypass the latter positive mass issue and show that any conformal infinity of a Poincaré-Einstein manifold of dimension either $n = 2$ or of dimension $n \geq 3$ and which is locally flat admits a Riemannian metric of constant fractional scalar curvature.

Fractional Yamabe problem on locally flat conformal infinities of Poincare-Einstein manifolds

Abstract

We study in this paper the fractional Yamabe problem first considered by Gonzalez-Qing on the conformal infinity of a Poincaré-Einstein manifold with either or and is locally flat - namely is locally conformally flat. However, as for the classical Yamabe problem, because of the involved quantization phenomena, the variational analysis of the fractional one exhibits also a local situation and a global one. Furthermore the latter global situation includes the case of conformal infinities of Poincaré-Einstein manifolds of dimension either 2 or of dimension greater than and which are locally flat, and hence the minimizing technique of Aubin- Schoen in that case clearly requires an analogue of the positive mass theorem of Schoen-Yau which is not known to hold. Using the algebraic topological argument of Bahri-Coron, we bypass the latter positive mass issue and show that any conformal infinity of a Poincaré-Einstein manifold of dimension either or of dimension and which is locally flat admits a Riemannian metric of constant fractional scalar curvature.

Paper Structure

This paper contains 14 sections, 15 theorems, 192 equations.

Table of Contents

  1. Introduction and statement of the results
  2. Preliminaries and notations
  3. Bubbles and related interaction estimates
  4. The bubbles on $\mathbb{R}^{n+1}_{+}$ as an interaction of standard ones on $\mathbb{R}^{n}$
  5. Schoen's bubbles and Projective bubbles
  6. Interaction estimates for $v_{a, \lambda }$ and related identities
  7. Locally flat conformal infinities of Poincaré-Einstein manifolds
  8. Fermi-coordinates in the particular case
  9. Comparing Schoen's bubbles and the Projective bubbles
  10. Selfaction estimates for $v_{a, \lambda }$ and the emergence of mass
  11. Variational and algebraic topological argument
  12. Variational principle in a non-compact setting
  13. Algebraic topological argument of Bahri-Coron
  14. Appendix

Key Result

Theorem 1.3

Let $n\geq 2$ be a positive integer, $(X^{n+1}, \; g^+)$ be a Poincaré-Einstein manifold with conformal infinity $(M^{n}, \;[h])$, $\gamma \in(0, 1)$, and $\left(\frac{n}{2}\right)^2-\gamma^2<\lambda _1(-\Delta _{g^+})$. Assuming that either $n=2$ or $n\ge 3$ and $(M, [h])$ is locally fl

Theorems & Definitions (22)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 3.1
  • Corollary 3.2
  • Corollary 3.3
  • Lemma 3.4
  • ...and 12 more