Communications in Analysis and Geometry

Volume 28 (2020)

Number 4

The First of Two Special Issues in Honor of Karen Uhlenbeck’s 75th Birthday

Special-Issue Editors: Georgios Daskalopoulos (Brown University), Kefeng Liu, Chuu-Lian Terng (U. of Cal. Irvine), and Shing-Tung Yau

Anti-self-dual $4$-manifolds, quasi-Fuchsian groups, and almost-Kähler geometry

Pages: 745 – 780

DOI: https://dx.doi.org/10.4310/CAG.2020.v28.n4.a1

Authors

Christopher J. Bishop (Department of Mathematics, Stony Brook University, Stony Brook, New York, U.S.A.)

Claude LeBrun (Department of Mathematics, Stony Brook University, Stony Brook, New York, U.S.A.)

Abstract

It is known that the almost-Kähler anti-self-dual metrics on a given $4$-manifold sweep out an open subset in the moduli space of antiself-dual metrics. However, we show here by example that this subset is not generally closed, and so need not sweep out entire connected components in the moduli space. Our construction hinges on an unexpected link between harmonic functions on certain hyperbolic $3$-manifolds and self-dual harmonic $2$-forms on associated $4$-manifolds.

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The first-named author was supported in part by NSF Grant DMS-1608577.

The second-named author was supported in part by NSF grant DMS-1510094.

Received 30 July 2017

Accepted 13 September 2017

Published 1 October 2020