Levi’s forms of higher codimensional submanifolds. (English) Zbl 0741.58045

D’Agnolo, Andrea; Zampieri, Giuseppe

Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 2, No. 1, 29-33 (1991).

It is proved that the number \(s(S,p)\), \(p\in\dot T^*_ SX\), defined as the Maslov index for three Lagrangian planes [see: M. Kashiwara and P. Schapira, ‘Microlocal study of sheaves’ (1985; Zbl 0589.32019)] coincides with the signature of the restriction on the complexified tangent space \(\mathbb{C} T_{z_ 0}S\), \(z_ 0\in S\), of the Levi form \(L_ M\), where \(M\) is a hypersurface of \(\mathbb{C}^ n\) and \(S\) is a submanifold of \(M\), i.e. \(s(S,p)=\hbox{sgn}(LM\mid \mathbb{C} T_{z_ 0}S)\). Obtaining a formula for the rank\((L_ M\mid \mathbb{C} T_{z_ 0}S)\), which includes the codimension of \(S\) in \(\mathbb{C}^ n\), the authors prove that, introduced by M. Kashiwara and P. Schapira [Invent. Math. 82, 579-592 (1985; Zbl 0626.58028)] the numbers \(s^ \pm(S,p)\) are respectively the positive and negative eigenvalues for \(L_ M\mid CT_{z_ 0}S\), which is of course true for \(S=M\). An application for microfunctions at the boundary are given.

Reviewer: S.Dimiev (Sofia)

MSC:

58J32	Boundary value problems on manifolds
58J15	Relations of PDEs on manifolds with hyperfunctions
35A27	Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
32C38	Sheaves of differential operators and their modules, \(D\)-modules

Keywords:

PDE on manifolds; boundary value problem; Levi form of submanifold

Citations:

Zbl 0589.32019; Zbl 0626.58028