The space of hyperk\"ahler metrics on a 4-manifold with boundary

TL;DR

This paper investigates the structure of the space of hyperk"ahler metrics on a 4-manifold with boundary, establishing its smoothness, describing its tangent space, and analyzing boundary value problems and explicit examples.

Contribution

It proves that the moduli space of hyperk"ahler triples on a 4-manifold with boundary is a smooth infinite-dimensional manifold and characterizes its tangent space and boundary deformations.

Findings

The moduli space is a smooth infinite-dimensional manifold.

The tangent space is described via triples of closed anti-self-dual 2-forms.

Explicit examples from gravitational instantons are analyzed.

Abstract

Let X be a compact 4-manifold with boundary. We study the space of hyperk\"ahler triples on X, modulo diffeomorphisms which are the identity on the boundary. We prove that this moduli space is a smooth infinite-dimensional manifold and describe the tangent space in terms of triples of closed anti-self-dual 2-forms. We also explore the corresponding boundary value problem: a hyperk\"ahler triple restricts to a closed framing of the bundle of 2-forms on the boundary; we identify the infinitesimal deformations of this closed framing that can be filled in to hyperk\"ahler deformations of the original triple. Finally we study explicit examples coming from gravitational instantons with isometric actions of SU(2).