A Conformally Invariant Classification Theorem in Four Dimensions

TL;DR

This paper establishes a classification theorem for 4-manifolds based on conformal invariants, extending previous results in conformal geometry and providing a four-dimensional analogue of a classical 3-manifold classification.

Contribution

It generalizes the conformally invariant sphere theorem and offers a new classification framework for 4-manifolds using conformal invariants.

Findings

Classification of 4-manifolds via conformal invariants

Extension of sphere theorem to four dimensions

Analogue of 3-manifold Yamabe classification in 4D

Abstract

In this paper, we prove a classification theorem of 4-manifolds according to some conformal invariants, which generalizes the conformally invariant sphere theorem of Chang-Gursky-Yang \cite{CGY}. Moreover, it provides a four-dimensional analogue of the well-known classification theorem of Schoen-Yau \cite{SY2} on 3-manifolds with positive Yamabe invariants.