Some conformally invariant gap theorems for Bach-flat 4-manifolds

TL;DR

This paper extends conformally invariant gap theorems for Bach-flat 4-manifolds beyond the sphere, including complex projective space and product of spheres, using iteration and convergence techniques.

Contribution

It generalizes previous results to new model cases, employing iteration and convergence methods for Bach-flat 4-manifolds.

Findings

Established gap theorems for $( ext{CP}^2, g_{FS})$ and $( ext{S}^2 imes ext{S}^2, g_{prod})$

Used iteration arguments for $( ext{CP}^2, g_{FS})$ case

Applied convergence theory for $( ext{S}^2 imes ext{S}^2, g_{prod})$ case

Abstract

Around 2007, A. Chang, J. Qing, and P. Yang proved a conformal gap theorem for Bach-flat metrics with round sphere as the model case. In this article, we extend this result to prove conformally invariant gap theorems for Bach-flat $4$-manifolds with $(\mathbb{CP}^2, g_{FS})$ and $(\mathbb{S}^2\times\mathbb{S}^2,g_{prod})$ as model cases. An iteration argument plays an important role in the case of $(\mathbb{CP}^2, g_{FS})$ and the convergence theory of Bach-flat metrics is of particular importance in the case of $(\mathbb{S}^2\times\mathbb{S}^2,g_{prod})$.