The large scale structure of complete $4$-manifolds with nonnegative Ricci curvature and Euclidean volume growth

TL;DR

This paper surveys the structure of 4-dimensional complete manifolds with nonnegative Ricci curvature and Euclidean volume growth, showing they resemble cones over spherical space forms at infinity, and discusses open questions and future directions.

Contribution

It provides a comprehensive survey of recent joint work on the asymptotic cone structure of such manifolds, highlighting new insights into their geometric behavior at infinity.

Findings

Manifolds resemble cones over spherical space forms at infinity

The structure of blow-downs reveals asymptotic geometric properties

Open questions suggest directions for future research

Abstract

We survey the implications of our joint work with E. Bru\`e and A. Pigati on the structure of blow-downs for a smooth, complete, Riemannian $4$-manifold with nonnegative Ricci curvature and Euclidean volume growth. Very imprecisely, any such manifold looks like a cone over a spherical space form at infinity. We present some open questions and discuss possible future directions along the way.