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Let E be a CM number field, F its maximal totally real subfield, c thegenerator of Gal(E/F), p an odd prime totally split in E, and S a finite set ofplaces of E containing the places above p.  Let r : G_{E,S} --> GL_3(F_p^bar) be a modular, absolutely irreducible,Galois representation of type U(3), i.e. such that r^* = r^c, and let X(r) bethe rigid analytic generic fiber of its universal G_{E,S}-deformation of typeU(3). We show that each irreducible component of the Zariski-closure of themodular points in X(r) has dimension at least 6[F:Q]. We study an analogue ofthe infinite fern of Gouvea-Mazur in this context and deal with the Hilbertmodular case as well.  As important steps, we prove that any first order deformation of a genericenough crystalline representation of Gal(Q_p^bar/Q_p) (of any dimension) is alinear combination of trianguline deformations, and that unitary eigenvarieties(of any rank) are etale over the weight space at the non-critical classicalpoints. As another application, we obtain a general theorem about the image ofthe localization at p of the p-adic Adjoint' Selmer group of the p-adic Galoisrepresentations attached to any cuspidal, cohomological, automorphicrepresentation Pi of GL_n(A_E) such that Pi^* = Pi^c (for any n).

On the infinite fern of Galois representations of unitary type