On stable singular Lagrangian varieties
type: Item
instance of: scholarly article
title: On stable singular Lagrangian varieties
zbMATH Open document ID: 0799.58033
DOI: 10.1007/BF01075630
publication date: 1994-07-06T00:00:00Z
review text: Singular Lagrangian varieties can be found in many problems of differential geometry, calculus of variations and mathematical physics. In Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Novejshie Dostizh. 33, 55-112 (1988; Zbl 0731.58023), \textit{A. B. Givental'} introduced the notion of stability of a singular Lagrangian germ with respect to perturbations of the symplectic structure and the Lagrangian projection. The present note contains a geometric interpretation of this notion for singular Lagrangian germs defined by generating families. The generating family is assumed to be equivalent to one induced from a versal deformation of a germ of a function with finite Milnor number. It turns out (cf. Theorems 7, 8) that the stability condition for such varieties imposes rather strong restrictions upon the inducing mappings of the parameter spaces. For example, a proper analytic map inducing a stable singular germ from a versal deformation of a simple singularity with a large number of parameters must be a rafimied covering of a stratum of the bifurcation diagram of functions for the singularity. Lagrangian germs associated with Coxeter groups arise naturally in the classification of simple stable regular inducing maps. In \S\S1 and 2 we show that the class of Lagrangian germs admitting a generating family is large enough and consider some properties of stable germs.
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 58E15
Mathematics Subject Classification ID: 32S30
zbMATH DE Number: 571842
zbMATH Keywords: singular Lagrangian varieties
zbMATH Keywords: stability
zbMATH Keywords: singularity
zbMATH Keywords: Lagrangian germs
zbMATH Keywords: Coxeter groups
author: V. M. Zakalyukin
author: R. M. Roberts
MaRDI profile type: Publication
cites work: Wavefronts and reflection groups
cites work: Generating families for images of Lagrangian submanifolds and open swallowtails
cites work: Q3357136
full work available at URL: https://doi.org/10.1007/bf01075630
OpenAlex ID: W2028903063
published in: Functional Analysis and Its Applications
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description: scientific article; zbMATH DE number 571842