Deformations of asymptotically cylindrical coassociative submanifolds with fixed boundary

Joyce, Dominic; Salur, Sema

doi:10.2140/gt.2005.9.1115

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Deformations of asymptotically cylindrical coassociative submanifolds with fixed boundary

Dominic Joyce and Sema Salur

Geometry & Topology 9 (2005) 1115–1146

DOI: 10.2140/gt.2005.9.1115

arXiv: math.DG/0408137

Abstract

McLean proved that the moduli space of coassociative deformations of a compact coassociative 4–submanifold $C$ in a $G_{2}$ –manifold $(M, φ, g)$ is a smooth manifold of dimension equal to $b_{+}^{2} (C)$ . In this paper, we show that the moduli space of coassociative deformations of a noncompact, asymptotically cylindrical coassociative 4–fold $C$ in an asymptotically cylindrical $G_{2}$ –manifold $(M, φ, g)$ is also a smooth manifold. Its dimension is the dimension of the positive subspace of the image of $H_{cs}^{2} (C, ℝ)$ in $H^{2} (C, ℝ)$ .

Keywords

calibrated geometries, asymptotically cylindrical manifolds, $G_2$–manifolds, coassociative submanifolds, elliptic operators.

Mathematical Subject Classification 2000

Primary: 53C38, 53C15, 53C21

Secondary: 58J05

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Publication

Received: 12 August 2004

Accepted: 7 May 2005

Published: 1 June 2005

Proposed: Rob Kirby

Seconded: Simon Donaldson, Gang Tian

Authors

Dominic Joyce
Lincoln College University of Oxford Oxford OX1 3DR United Kingdom

Sema Salur
Department of Mathematics Northwestern University Illinois 60208 USA

Volume 9, issue 2 (2005)