  We prove the existence of non-classical $p$-adic automorphic eigenforms
associated to a classical system of eigenvalues on definite unitary groups in
$3$ variables. These eigenforms are associated to Galois representations which
are crystalline but very critical at $p$. We use patching techniques related to
the trianguline variety of local Galois representations and its local model.
The new input is a comparison of the coherent sheaves appearing in the patching
process with coherent sheaves on the Grothendieck--Springer version of the
Steinberg variety given by a functor constructed by Bezrukavnikov.
