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Perturbation by analytic discs along maximal real submanifolds of \(C^ N\). (English) Zbl 0806.58044
Let \(\Delta\) be the open unit disc in \(\mathbb{C}\). Given a totally real submanifold \(M\) of \(\mathbb{C}^ N\) of dimension \(N\) and a smooth map \(p : b \Delta \to M\); we want to find smooth maps \(\varphi : \overline{\Delta} \to C^ N\), holomorphic on \(\Delta\) which are close to the zero map and which satisfy \((p + \varphi) (b\Delta) \subset M\). For each \(\zeta \in b\Delta\) let \(T(\zeta)\) be the tangent space to \(M\) at \(p(\zeta)\). In the cases considered the bundle \(T = \{T(\zeta) : \zeta \in b\Delta\}\) is trivial. We introduce the partial indices of \(M\) along \(p\) as the partial indices of a vector Hilbert problem naturally associated with the bundle \(T\), and the total index \(\kappa\) of \(M\) along \(p\) which is the sum of the partial indices. We show that if all partial indices of \(M\) along \(p\) are nonnegative then the family of maps \(\varphi\) above depends on \(\kappa + N\) real parameters and this is still true for small perturbations of \(M\). This generalizes a result of Forstnerič who studied the case when \(N = 2\) and when \(p = f\mid b \Delta\) where \(f: \overline{\Delta} \to \mathbb{C}^ 2\) is an immersed analytic disc [F. Forstnerič, Ann. Inst. Fourier 37, 1-44 (1987; Zbl 0583.32038)].

MSC:
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
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