Aubin’s Lemma for the Yamabe constants of infinite coverings and a positive mass theorem

Aubin’s Lemma says that, if the Yamabe constant of a closed conformal manifold (M, C) is positive, then it is strictly less than the Yamabe constant of any of its non-trivial finite conformal coverings. We generalize this lemma to the one for the Yamabe constant of any (M∞, C∞) of its infinite conformal coverings, provided that π1(M) has a descending chain of finite index subgroups tending to π1(M∞). Moreover, if the covering M∞ is normal, the limit of the Yamabe constants of the finite conformal coverings (associated to the descending chain) is equal to that of (M∞, C∞). For the proof of this, we also establish a version of positive mass theorem for a specific class of asymptotically flat manifolds with singularities.