Addendum to: ``Subelliptic Spin\(_{\mathbb{C}}\) Dirac operators. III''.
type: Item
instance of: scholarly article
title: Addendum to: ``Subelliptic Spin\(_{\mathbb{C}}\) Dirac operators. III''.
zbMATH Open document ID: 1256.32034
author: Charles L. Epstein
published in: Annals of Mathematics. Second Series
publication date: 2012-10-16T00:00:00Z
Mathematics Subject Classification ID: 32V30
Mathematics Subject Classification ID: 32W10
Mathematics Subject Classification ID: 32V15
zbMATH DE Number: 6093951
zbMATH Keywords: contact manifold
zbMATH Keywords: embeddable CR-structure
zbMATH Keywords: relative index conjecture
zbMATH Keywords: Stein filling
zbMATH Keywords: strictly pseudoconvex surface
reviewed by: Jiri Lebl
MaRDI profile type: Publication
full work available at URL: annals.2012.176.2.12
OpenAlex ID: W2109462767
cites work: Q4355863
cites work: A relative index on the space of embeddable CR-structures. I
cites work: Erratum: A relative index on the space of embeddable CR-structures. I
cites work: A relative index on the space of embedabble CR-structures. II
cites work: Subelliptic Spin\(_\sigma\) Dirac operators. III: The Atiyah-Weinstein conjecture
cites work: On Three-Dimensional Cauchy-Riemann Manifolds
cites work: Embedding of three dimensional Cauchy-Riemann manifolds
cites work: Algebraic approximations in analytic geometry
cites work: On the geography of Stein fillings of certain 3-manifolds
DOI: 10.4007/ANNALS.2012.176.2.12
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review text: In this addendum to [Ann. Math. (2) 168, No. 1, 299--365 (2008; Zbl 1169.32008)] the author proves the relative index conjecture. Using this result the author proves that the set of embeddable deformations of a strictly pseudoconvex CR-structure on a compact 3-manifold is closed in the \(C^\infty\)-topology.NEWLINENEWLINEIn particular, let \(Y\) be a 3 dimensional manifold and \(H \subset TY\) a plane field defining a contact structure. We can define a strictly pseudoconvex CR-structure on \(Y\) by a complex structure on \(H\). The manifold is fillable if it can be embedded into some \({\mathbb{C}}^N\) as a boundary of a normal Stein space. A Szegő projector is the \(L^2\)-orthogonal projection onto the \(L^2\)-closure of \(\ker \bar{\partial}_b\). If \(S_1\) is another Szegő projector defined by a fillable deformation of the reference structure then the restriction \(S_1 : \text{Im} S_0 \to \text{Im} S_1\) is a Fredholm operator and the Fredholm index is the relative index \(\text{R-Ind}(S_0,S_1)\). See also [the author, Ann. Math. (2) 147, No. 1, 1--59 (1998; Zbl 0942.32025)].NEWLINENEWLINEIn this addendum the author proves that if \((Y,H)\) is a compact 3-dimensional co-oriented contact manifold and \(S_0\) the Szegő projector defined by a fillable CR-structure on \(Y\) with underlying plane field \(H\), then there is an \(M \geq 0\) such that if \(S_1\) is the Szegő projector defined by any fillable deformation of the reference structure, with the plane field \(H\), then \(\text{R-Ind}(S_0,S_1) \leq M\). As a corollary, the set of fillable deformations of the CR structure on \(Y\) is closed in the \(C^\infty\)-topology.
description: scientific article; zbMATH DE number 6093951