On stable singular Lagrangian varieties
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.
scientific article; zbMATH DE number 571842
scientific article