Uniformization theorems: Between Yamabe and Paneitz

Cheikh Birahim Ndiaye, Yannick Sire and Liming Sun

Vol. 314 (2021), No. 1, 115–159

DOI: 10.2140/pjm.2021.314.115

Abstract

This paper is devoted to several existence results for a generalized version of the Yamabe problem. First, we prove the remaining global cases for the range of powers $\gamma \in \left(0,1\right)$ for the generalized Yamabe problem introduced by Gonzalez and Qing. Second, building on a new approach by Case and Chang for this problem, we prove that this Yamabe problem is solvable in the Poincaré-Einstein case for $\gamma \in \left(1,min\left\{2,\frac{n}{2}\right\}\right)$ provided the associated fractional GJMS operator satisfies the strong maximum principle.

Keywords

fractional GJMS operator, Poincaré–Einstein manifold, algebraic topological argument, barycenter technique

Mathematical Subject Classification

Primary: 58J05

Secondary: 35R11, 53A31

Milestones

Received: 24 July 2020

Revised: 24 May 2021

Accepted: 9 July 2021

Published: 15 October 2021