Communications in Analysis and Geometry
Volume 28 (2020)
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
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.
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