  Let $F/F^+$ be a CM extension and $H_{/F^+}$ a definite unitary group in
three variables that splits over $F$. We describe Hecke isotypic components of
mod $p$ algebraic modular forms on $H$ at first principal congruence level at
$p$ and "minimal" level away from $p$ in terms of the restrictions of the
associated Galois representation to decomposition groups at $p$ when these
restrictions are tame and sufficiently generic. This confirms an expectation of
local-global compatibility in the mod $p$ Langlands program. To prove our
result, we develop a local model theory for multitype deformation rings and new
methods to work with patched modules that are not free over their
scheme-theoretic support.
