  Conjecturally, the Galois representations that are attached to essentially
selfdual regular algebraic cuspidal automorphic representations are
Zariski-dense in a polarized Galois deformation ring. We prove new results in
this direction in the context of automorphic forms on definite unitary groups
over totally real fields. This generalizes the infinite fern argument of
Gouvea-Mazur and Chenevier, and relies on the construction of non-classical
$p$-adic automorphic forms, and the computation of the tangent space of the
space of trianguline Galois representations. This boils down to a surprising
statement about the linear envelope of intersections of Borel subalgebras.
