  In Euclidean 3-space endowed with a Cartesian reference system we consider a
class of surfaces, called Delaunay tori, constructed by bending segments of
Delaunay cylinders with neck-size $a$ and $n$ lobes along circumferences
centered at the origin. Such surfaces are complete and compact, have genus one
and almost constant, say 1, mean curvature, when $n$ is large. Considering a
class of mappings $H\colon\mathbb{R}^{3}\to\mathbb{R}$ such that $H(X)\to 1$ as
$|X|\to\infty$ with some decay of inverse-power type, we show that for $n$
large and $|a|$ small, in a suitable neighborhood of any Delaunay torus with
$n$ lobes and neck-size $a$ there is no parametric surface constructed as
normal graph over the Delaunay torus and whose mean curvature equals $H$ at
every point.
