  The properties of the Gribov region in SU(2) Euclidean Yang-Mills theories in
the maximal Abelian gauge are investigated. This region turns out to be bounded
in all off-diagonal directions, while it is unbounded along the diagonal one.
The soft breaking of the BRST invariance due to the restriction of the domain
of integration in the path integral to the Gribov region is scrutinized. Owing
to the unboundedness in the diagonal direction, the invariance with respect to
Abelian transformations is preserved, a property which is at the origin of the
local U(1) Ward identity of the maximal Abelian gauge.

High Energy Physics - Theory