Continued fractions and Einstein manifolds of infinite topological type

TL;DR

This paper constructs a vast family of complete self-dual Einstein metrics with negative scalar curvature on infinitely complex topological manifolds, using continued fractions to parameterize the manifolds.

Contribution

It introduces a novel method to generate infinite topological type Einstein manifolds via continued fraction expansions, extending previous rational cases.

Findings

Constructed uncountably many such manifolds

Connected continued fractions to geometric structures

Extended previous rational quotient singularity resolutions

Abstract

We present a construction of complete self-dual Einstein metrics of negative scalar curvature on an uncountable family of manifolds of infinite topological type, which are enumerated by continued fraction expansions of irrational numbers. These manifolds may be regarded as limits of the resolutions of cyclic quotient singularities (governed by continued fraction expansions of rational numbers) on which we constructed complete self-dual Einstein metrics in previous work.