Rigidity of Schouten Tensor under Conformal Deformation

TL;DR

This paper proves rigidity results for metrics with Schouten tensor bounded below after conformal changes, providing a simple proof of Cheng's theorem on locally conformally flat manifolds with Ricci pinching.

Contribution

It establishes new rigidity results for conformally deformed metrics with Schouten tensor bounds, simplifying proofs of existing theorems.

Findings

Rigidity results for metrics with Schouten tensor bounds

A simple proof of Cheng's theorem on Ricci pinching

Confirmation of Hamilton's pinching conjecture in higher dimensions

Abstract

We obtain some rigidity results for metrics whose Schouten tensor is bounded from below after conformal transformations. Liang Cheng recently proved that a complete, nonflat, locally conformally flat manifold with Ricci pinching condition ($Ric-\epsilon Rg\geq 0$) must be compact. This answers higher dimensional Hamilton's pinching conjecture on locally conformally flat manifolds affirmatively. Since (modified) Schouten tensor being nonnegative is equivalent to a Ricci pinching condition, our main result yields a simple proof of Cheng's theorem.