On the ADM mass of K\"ahler scalar flat ALE metrics

TL;DR

This paper investigates how the ADM mass of scalar flat Kähler ALE spaces changes under blow ups, demonstrating that sufficiently many blow ups can produce arbitrarily large ADM mass and presenting a method to construct spaces with zero ADM mass.

Contribution

It introduces a new approach to analyze the ADM mass behavior under blow ups and provides a general method to construct scalar flat ALE spaces with specified ADM mass.

Findings

Blowing up points at large mutual distances increases ADM mass.

Sufficiently many blow ups can produce arbitrarily large ADM mass.

A machinery for constructing scalar flat ALE spaces with zero ADM mass is developed.

Abstract

In this paper we study the behaviour of scalar flat K\"ahler ALE spaces and their ADM mass under blow ups. In particular we prove that by blowing up sufficiently many points at sufficiently big mutual distance one can produce scalar flat metrics with arbitrarily large ADM mass. A general machinery for producing scalar flat non Ricci flat ALE spaces of zero ADM mass is also presented, using and integrating previous work by Rollin-Singer and Hein-LeBrun.