Mass Lower Bounds for Asymptotically Locally Flat Manifolds

Abstract

We establish positive mass type theorems for asymptotically locally flat (ALF) manifolds, which have asymptotic ends modeled on circle bundles over a Euclidean base with fibers of constant length. In particular for dimensions $n\leq 7$, the mass of AF manifolds is shown to be nonnegative under the assumption of nonnegative scalar curvature if a codimension-two coordinate sphere in the asymptotic end is trivial in homology, with zero mass achieved only for the product $\mathbb{R}^{n-1}\times S^1$. The same conclusions are obtained in dimension four for ALF manifolds admitting an almost free $U(1)$ action. Moreover, in this setting the mass is shown to be bounded below by a multiple of the degree of the circle bundle at infinity. This is the first such result illustrating how nontrivial topology of the end contributes to the mass.