Uniformization Theorems: Between Yamabe and Paneitz

TL;DR

This paper advances the understanding of generalized Yamabe problems by establishing new existence results across different parameter ranges and geometric contexts, including Poincaré-Einstein manifolds, using novel analytical techniques.

Contribution

It proves the remaining global cases for the generalized Yamabe problem with powers in (0,1) and solves the problem in Poincaré-Einstein cases for powers in (1, min{2,n/2}) under certain operator conditions.

Findings

Established existence for $eta o 0$ in the generalized Yamabe problem.

Proved solvability in Poincaré-Einstein manifolds for specific fractional powers.

Extended the range of parameters for which the Yamabe problem is solvable.

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 $\gamma\in (0,1)$ 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 Poincar\'{e}-Einstein case for $\gamma\in (1,\min\{2,n/2\})$ provided the associated fractional GJMS operator satisfies the strong maximum principle.