# Almost non-negatively curved 4-manifolds with torus symmetry

## TL;DR

This paper demonstrates that certain 4-manifolds with torus symmetry that admit almost non-negative curvature metrics can be equipped with genuinely non-negative curvature metrics, leading to a classification of such manifolds.

## Contribution

It establishes that almost non-negatively curved 4-manifolds with torus symmetry can be smoothed to non-negative curvature, extending classification results in this geometric setting.

## Key findings

- Almost non-negatively curved 4-manifolds with torus symmetry admit non-negative curvature metrics.
- Classification of simply-connected 4-manifolds with torus symmetry under almost non-negative curvature.
- Extension of curvature smoothing techniques to higher-rank torus actions.

## Abstract

We prove that if a closed, smooth, simply-connected 4-manifold with a circle action admits an almost non-negatively curved sequence of invariant Riemannian metrics, then it also admits a non-negatively curved Riemannian metric invariant with respect to the same action. The same is shown for torus actions of higher rank, giving a classification of closed, smooth, simply-connected 4-manifolds of almost non-negative curvature under the assumption of torus symmetry.