On the conformal bending of a closed Riemannian manifold

TL;DR

This paper demonstrates that any closed Riemannian manifold with quasi-negative Ricci curvature can be conformally transformed into a metric with negative Ricci curvature by solving a fully nonlinear equation.

Contribution

It introduces a method to conformally deform metrics within a class to achieve negative Ricci curvature, solving a fully nonlinear PDE.

Findings

Any quasi-negative Ricci curvature metric is conformal to a negative Ricci curvature metric.

The conformal deformation is achieved by solving a specific nonlinear PDE.

The approach applies to all closed Riemannian manifolds with quasi-negative Ricci curvature.

Abstract

In this paper, we bend a closed Riemannian manifold in the conformal class, through solving a fully nonlinear equation. As a result, we prove that each metric of quasi-negative Ricci curvature is conformal to a metric with negative Ricci curvature.