Gap Theorems for Locally Conformally Flat Manifolds

TL;DR

This paper establishes a gap theorem for certain non-compact, locally conformally flat manifolds with specific curvature conditions, showing they must be flat if they have a positive Green function, using a new global Yamabe flow approach.

Contribution

It introduces a novel gap theorem for locally conformally flat manifolds under curvature and scalar conditions, utilizing a new global Yamabe flow method.

Findings

Manifolds with positive Green function are flat under given conditions

New global Yamabe flow technique developed for the proof

Extensions to solutions of a Schrödinger equation discussed

Abstract

In this paper, we prove a gap result for a locally conformally flat complete non-compact Riemannian manifold with bounded non-negative Ricci curvature and a scalar curvature average condition. We show that if it has positive Green function, then it is flat. This result is proved by setting up new global Yamabe flow. Other extensions related to bounded positive solutions to a schrodinger equation are also discussed.