Density of automorphic points in deformation rings of polarized global Galois representations

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.