The ABP method for proving isoperimetric inequalities has been first employed by Cabré in \({{\,{\mathbb {R}}\,}}^n,\) then developed by Brendle, notably in the context of non-compact Riemannian manifolds of non-negative Ricci curvature and positive asymptotic volume ratio. In this paper, we expand upon their approach and prove isoperimetric inequalities (sharp in the limit) in the presence of a small amount of negative curvature. First, we consider smallness of the negative part \(\text{Ric}_-\) of the Ricci curvature in terms of its elliptic Kato constant. Indeed, the Kato constant turns out to control the non-negativity of the ( \(\infty\) )- Bakry-Émery Ricci-tensor of a suitable conformal deformation of the manifold, and the ABP method can be implemented in this setting. Secondly, we show that the smallness of the Kato constant is ensured provided that the asymptotic volume ratio is positive and either M has one end and asymptotically non-negative sectional curvature, or there is a suitable polynomial decay of \(\text{Ric}_{-},\) and the relative volume comparison condition known as \({\textbf {(VC)}}\) holds. To show this latter fact, we enhance techniques elaborated by Li-Tam and Kasue to obtain new estimates of the Green function valid on the whole manifold.
