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THE SPACE OF HYPERKÄHLER METRICS ON A 4-MANIFOLD WITH BOUNDARY

Published online by Cambridge University Press:  01 March 2017

JOEL FINE
Affiliation:
Départment de mathématique, Université libre de Bruxelles (ULB), CP 218, Boulevard du Triomphe, B-1050 Bruxelles, Belgium; joel.fine@ulb.ac.be
JASON D. LOTAY
Affiliation:
Department of Mathematics, University College, London WC1E 6BT, UK; j.lotay@ucl.ac.uk, michael.singer@ucl.ac.uk
MICHAEL SINGER
Affiliation:
Department of Mathematics, University College, London WC1E 6BT, UK; j.lotay@ucl.ac.uk, michael.singer@ucl.ac.uk

Abstract

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Let $X$ be a compact 4-manifold with boundary. We study the space of hyperkähler triples $\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}_{2},\unicode[STIX]{x1D714}_{3}$ 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 hyperkähler 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 hyperkähler deformations of the original triple. Finally we study explicit examples coming from gravitational instantons with isometric actions of $\text{SU}(2)$.

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2017

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