  To each complex saddle point of an action, one can attach a Lefschetz thimble
on which the imaginary part of the action is constant. Cauchy theorem states
that summation over a set of thimbles produces the exact result. This
reorganization of the path integral, is an appealing starting point for various
approximations: In the realm of auxiliary quantum Monte Carlo methods it
provides a framework to alleviate the negative sign problem. Here, we suggest
to constrain the integration to the \textit{dominant} thimbles: the thimbles
attached to the saddle points with the largest statistical weight. For the
Hubbard model, in a formulation where the the Hubbard Stratonovitch field
couples to the charge, this provides a \textit{symmetry} consistent
approximation to the physics of the Hubbard model: constraining the integration
domain does not explicitly break a symmetry. We can test this approach for the
Hubbard model at half-filling on a bipartite lattice. The paper builds on the
previously developed instanton gas approach, where an exhaustive saddle point
approximation was constructed. We present results, showing that the dominant
thimbles approximation provides results that are in remarkable agreement with
the exact results for various fermionic observables including spin and charge
order parameters and single electron spectral functions. We discuss
implications of our results for simulations away of half filling.
