Gravitational instantons and harmonic maps

TL;DR

This paper explores the relationship between toric Ricci-flat metrics in four dimensions and axisymmetric harmonic maps, leading to new constructions of Ricci-flat manifolds and classifications of harmonic maps with applications in gravitational instantons.

Contribution

It introduces a non-perturbative method for constructing Ricci-flat 4-manifolds and classifies axisymmetric harmonic maps of degree at most 1 using the Gibbons-Hawking and LeBrun-Tod ansatz.

Findings

Constructed complete Ricci-flat 4-manifolds with arbitrary second Betti number.

Provided counterexamples to Riemannian black hole uniqueness conjecture.

Classified low-degree axisymmetric harmonic maps via PDE analysis.

Abstract

We study the interaction between toric Ricci-flat metrics in dimension 4 and axisymmetric harmonic maps from the 3-dimensional Euclidean space into the hyperbolic plane. Applications include (1). The construction of complete Ricci-flat 4-manifolds that are non-spin, simply-connected, and with arbitrary second Betti number. Our method is non-perturbative and is based on ruling out conical singularities arising from axisymmetric harmonic maps. These metrics also give systematic counterexamples to various versions of the Riemannian black hole uniqueness conjecture. (2). A PDE classification result for axisymmetric harmonic maps of degree at most 1, via the Gibbons-Hawking ansatz and the LeBrun-Tod ansatz, in terms of axisymmetric harmonic functions. This is motivated by the study of hyperkahler and conformally Kahler gravitational instantons.