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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.

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