Notes on the uniqueness of Type II Yamabe metrics

TL;DR

This paper investigates the uniqueness of Type II Yamabe metrics on manifolds with boundary and extends Obata-type theorems to these metrics and the CR Yamabe problem, providing conditions for their uniqueness.

Contribution

It establishes new sufficient conditions for the uniqueness of Type II Yamabe metrics and extends Obata-type theorems to the CR Yamabe setting.

Findings

Provided a theorem for the uniqueness of Type II Yamabe metrics

Extended Obata-type theorems to boundary and CR cases

Identified conditions ensuring metric uniqueness

Abstract

In this paper, we study the uniqueness of type II Yamabe metrics in conformal classes on a compact connected manifold with boundary, and we investigate Obata-type theorems for type II Yamabe metrics. In particular, we establish a theorem which gives a sufficient condition for a metric to be the unique Type II Yamabe metric in its conformal class. We also prove the corresponding theorem for the CR Yamabe problem on closed manifolds.