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

Abstract

We survey the implications of our joint work with E. Bruè 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.

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