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

We show that if a complete Riemannian $3-$manifold has $L^{\frac{3}{2}}-$ integrable Ricci curvature, satisfies a Sobolev inequality and has a non negative Ricci curvature in a spectral sense, then it is diffeomorphic to ${\mathbb{R}}^3$.