# The renormalized volume of a 4-dimensional Ricci-flat ALE space

## TL;DR

This paper defines and analyzes the renormalized volume of 4-dimensional Ricci-flat ALE spaces, proving it is non-positive and characterizing cases of equality, with explicit calculations for known examples.

## Contribution

It introduces a natural definition of renormalized volume for these spaces and establishes a key inequality with equality characterization, linking to known examples.

## Key findings

- Renormalized volume is always ≤ 0 for these spaces.
- Equality holds if and only if the space is isometric to its asymptotic cone.
- Explicit calculations for Kronheimer's gravitational instantons.

## Abstract

We introduce a natural definition of the renormalized volume of a 4-dimensional Ricci-flat ALE space. We then prove that the renormalized volume is always less or equal than zero, with equality if and only if the ALE space is isometric to its asymptotic cone. Currently the only known examples of 4-dimensional Ricci-flat ALE spaces are Kronheimer's gravitational instantons and their quotients, which are also known to be the only possible examples of special holonomy. We calculate the renormalized volume of these spaces in terms of Kronheimer's period map.