Open $3$-manifolds with non negative Ricci curvature in a spectral or integral sense

TL;DR

This paper proves that complete 3-manifolds with certain integral Ricci curvature bounds, Sobolev inequalities, and spectral non-negativity are topologically equivalent to Euclidean space.

Contribution

It establishes a new topological classification for 3-manifolds under integral Ricci curvature and spectral conditions, extending previous results.

Findings

Manifolds with $L^{3/2}$ Ricci integrability are diffeomorphic to $ ^3$

Sobolev inequalities combined with spectral Ricci conditions imply Euclidean topology

Non-negative spectral Ricci curvature influences manifold topology significantly.

Abstract

We show that if a complete Riemannian $3-$manifold has $L^{\frac 32}-$ integrable Ricci curvature, satisfies a Sobolev inequality and has a non negative Ricci curvature in a spectral sense, then it is diffeomorphic to $\R^3$.