Mass and Expansion of Asymptotically Conical K\"ahler Metrics

TL;DR

This paper proves an expansion theorem for scalar-flat asymptotically conical K"ahler metrics, showing their leading behavior is determined by the cone metric and relating the mass to the metric's asymptotics, with applications to positive mass theorems.

Contribution

It establishes an expansion theorem for scalar-flat AC K"ahler metrics and extends the mass formula and positive mass theorem to this setting.

Findings

Scalar-flat AC K"ahler metrics admit an explicit asymptotic expansion.

The leading error term is of order O(r^{2-2n}) and depends on the ADM mass.

The mass formula by Hein-LeBrun is valid in this context.

Abstract

We prove an expansion theorem on scalar-flat asymptotically conical K\"ahler metrics. Consider an AC K\"ahler manifold with asymptotic to a Ricci-flat K\"ahler metric cone with complex dimension n. Assuming the weak decay conditions required for the mass to be well-defined, then each scalar-flat AC K\"ahler metric admits an expansion that the main term is given by the standard K\"ahler metric of the metric cone and the leading error term is of O(r^{2-2n}) with coefficient only depending on the ADM mass and its dimension. Besides, the mass formula by Hein-LeBrun also can be proved in our setting. As an application, a new version of the positive mass theorem will be discussed in the cases of the resolutions of the Ricci-flat K\"ahler cones.