Volume 10, issue 3 (2006)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Obstructions to special Lagrangian desingularizations and the Lagrangian prescribed boundary problem

Mark Haskins and Tommaso Pacini

Geometry & Topology 10 (2006) 1453–1521

arXiv: math.DG/0609352

Abstract

We exhibit infinitely many, explicit special Lagrangian isolated singularities that admit no asymptotically conical special Lagrangian smoothings. The existence/ nonexistence of such smoothings of special Lagrangian cones is an important component of the current efforts to understand which singular special Lagrangians arise as limits of smooth special Lagrangians.

We also use soft methods from symplectic geometry (the relative version of the h–principle for Lagrangian immersions) and tools from algebraic topology to prove (both positive and negative) results about Lagrangian desingularizations of Lagrangian submanifolds with isolated singularities; we view the (Maslov-zero) Lagrangian desingularization problem as the natural soft analogue of the special Lagrangian smoothing problem.

Keywords
singular special Lagrangian submanifolds, smoothing, Lagrangian h–principle, conical singularities
Mathematical Subject Classification 2000
Primary: 53D12, 53C38
Secondary: 53C42
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Publication
Received: 2 July 2006
Accepted: 6 September 2006
Published: 21 October 2006
Proposed: Frances Kirwan
Seconded: Jim Bryan, Simon Donaldson
Authors
Mark Haskins
Department of Mathematics
Imperial College London
South Kensington
London SW7 2AZ
England
Tommaso Pacini
Georgia Institute of Technology
Department of Mathematics
686 Cherry Street
Atlanta, GA 30332
USA