# On 3-manifolds with pointwise pinched nonnegative Ricci curvature

## TL;DR

This paper proves that complete 3-manifolds with bounded sectional curvature and pointwise pinched nonnegative Ricci curvature are flat or compact, under the condition that the negative sectional curvature decays quadratically.

## Contribution

It establishes the conjecture for 3-manifolds with quadratic decay of negative sectional curvature, extending previous results in geometric analysis.

## Key findings

- Manifolds are flat or compact under the given conditions
- Quadratic decay of negative curvature is sufficient for the conjecture
- Supports the conjecture in a broader class of 3-manifolds

## Abstract

There is a conjecture that a complete Riemannian 3-manifold with bounded sectional curvature, and pointwise pinched nonnegative Ricci curvature, must be flat or compact. We show that this is true when the negative part (if any) of the sectional curvature decays quadratically.