2017/04/09 by Stefan Müller, Müller, Stefan
Mathematics · #53D10 #53D12 #58C10 #FOS: Mathematics #Symplectic Geometry (math.SG) #math.SG #msc:53D10 #msc:53D12 #msc:58C10
paper · pdf · doi:10.48550/arxiv.1704.02655
13 pages; v2: fixed technical issue regarding the function Lambda in (the old) subsection 4.2; most changes in (new) subsections 4.2 and 4.3 (up to end of proof of lemma), and minor cosmetic changes and typos corrected throughout; none of results affected by update
arxiv created 2017/04/28 · arxiv updated 2017/05/02
The purpose of this paper is to carry out a classical construction of a non-constant holomorphic disk with boundary on (the suspension of) a Lagrangian submanifold in ℝ2 n in the case the Lagrangian is the lift of a coisotropic (a.k.a. pre-Lagrangian) submanifold in (a subset U of) ℝ2 n - 1. We show that the positive lower and finite upper bounds for the area of such a disk (which are due to M. Gromov and J.-C. Sikorav and F. Laudenbach-Sikorav for general Lagrangians) depend on the coisotropic submanifold only but not on its lift to the symplectization. The main application is to a C0-characterization of contact embeddings in terms of coisotropic embeddings in another paper by the present author. Moreover, we prove a version of Gromov's non-existence of exact Lagrangian embeddings into standard ℝ2 n for coisotropic embeddings into S1 × ℝ2 n. This allows us to distinguish different contact structures on the latter by means of the (modified) contact shape invariant. As in the general Lagrangian case, all of the existence results are based on Gromov's theory of J-holomorphic curves and his compactness theorem (or persistence principle). Analytical difficulties arise mainly at the ends of the cone ℝ+ × U.