A homogeneous Gibbons-Hawking ansatz and Blaschke products

TL;DR

This paper introduces a homogeneous Gibbons-Hawking ansatz combined with Blaschke products to construct infinite families of hyperkahler metrics, revealing new geometric structures and properties in differential geometry.

Contribution

It presents a novel method to generate infinite-dimensional hyperkahler metrics using a homogeneous Gibbons-Hawking ansatz and Blaschke products, expanding the toolkit for geometric analysis.

Findings

Constructed infinite-dimensional families of hyperkahler metrics.

Produced incomplete metrics on 3D contact manifolds with complete Carnot-Caratheodory distances.

Demonstrated the interplay between complex analysis and differential geometry in metric construction.

Abstract

A homogeneous Gibbons-Hawking ansatz is described, leading to 4-dimensional hyperkahler metrics with homotheties. In combination with Blaschke products on the unit disc in the complex plane, this ansatz allows one to construct infinite-dimensional families of such hyperkahler metrics that are, in a suitable sense, complete. Our construction also gives rise to incomplete metrics on 3-dimensional contact manifolds that induce complete Carnot-Caratheodory distances.