Volume 23, issue 7 (2019)

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The classification of Lagrangians nearby the Whitney immersion

Georgios Dimitroglou Rizell

Geometry & Topology 23 (2019) 3367–3458
Abstract

The Whitney immersion is a Lagrangian sphere inside the four-dimensional symplectic vector space which has a single transverse double point of Whitney self-intersection number + 1. This Lagrangian also arises as the Weinstein skeleton of the complement of a binodal cubic curve inside the projective plane, and the latter Weinstein manifold is thus the “standard” neighbourhood of Lagrangian immersions of this type. We classify the Lagrangians inside such a neighbourhood which are homologically essential, and which are either embedded or immersed with a single double point; they are shown to be Hamiltonian isotopic to either product tori, Chekanov tori, or rescalings of the Whitney immersion.

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Keywords
nearby Lagrangian conjecture, Lagrangian fibration, Clifford torus, Chekanov torus, Whitney immersion, Whitney sphere
Mathematical Subject Classification 2010
Primary: 53D12
References
Publication
Received: 5 March 2018
Revised: 23 October 2018
Accepted: 7 March 2019
Published: 30 December 2019
Proposed: Yasha Eliashberg
Seconded: Ciprian Manolescu, Leonid Polterovich
Authors
Georgios Dimitroglou Rizell
Department of Mathematics
Uppsala University
Uppsala
Sweden