Energy and Asymptotics of Ricci-Flat 4-Manifolds with a Killing Field

TL;DR

This paper investigates Ricci-flat 4-manifolds with a Killing field, providing energy estimates based on asymptotic properties and establishing flatness under specific conditions.

Contribution

It introduces new energy bounds for Ricci-flat 4-manifolds with Killing fields and characterizes flatness when the Killing field is non-vanishing and asymptotically constrained.

Findings

Energy of the manifold is bounded by asymptotic data of the Killing field.

Manifolds are flat if the Killing field has no zeros and satisfies certain asymptotic conditions.

Provides criteria for flatness based on Killing field properties.

Abstract

Given a complete, Ricci-flat 4-manifold with a Killing field, we give an estimate on the manifold's energy in terms of a certain asymptotic quantity of the Killing field. If the Killing field has no zeros and satisfies a certain asymptotic condition, the manifold is flat.