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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
##### Keywords:
analytic discs; real submanifold; perturbations
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##### References:
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