A Comparison Theorem For the Mass of ALE and ALF Toric 4-Manifolds

TL;DR

This paper establishes sharp lower bounds for the mass of ALE and ALF toric 4-manifolds using gravitational instantons and geometric defects, with rigidity results characterizing equality cases.

Contribution

It introduces a new lower bound for the mass of ALE/ALF toric 4-manifolds in terms of gravitational instantons and conical defects, extending positive mass theorems.

Findings

Lower bounds for mass in terms of gravitational instantons.

Rigidity results when bounds are saturated.

Generalization to manifolds with singularities.

Abstract

We establish sharp lower bounds for the mass of asymptotically locally Euclidean (ALE) and asymptotically locally flat (ALF) toric 4-manifolds, in terms of equilibrium geometries consisting of gravitational instantons. More precisely, the mass of a complete ALE or ALF toric 4-manifold with nonnegative scalar curvature is bounded below by a sum comprised of the following quantities: the mass of the corresponding toric gravitational instanton having the same orbit space (rod) structure as the original ALE/ALF manifold, and an expression determined by the conical angle defects of totally geodesic 2-spheres within the instanton that serve as generators for its second homology. The inequality may be generalized to the situation in which the ALE/ALF manifold also possesses conical singularities as well as orbifold singularities, and it suggests a refined notion of `total mass' in which the result simply states that the total mass of the ALE/ALF manifold is not less than that of the corresponding gravitational instanton. Furthermore, we prove rigidity for these statements, namely the inequality is saturated only when the ALE/ALF manifold is Ricci flat and in fact agrees with the corresponding instanton. These results may be viewed in the context of positive mass theorems, providing an explanation of how positivity can fail in the ALE/ALF setting. Moreover, the main theorem may be interpreted as yielding a variational characterization of the relevant toric gravitational instantons.