  We construct a sequence of compact, oriented, embedded, two-dimensional
surfaces of genus one into Euclidean 3-space with prescribed, almost constant,
mean curvature of the form $H(X)=1+{A}{|X|^{-\gamma}}$ for $|X|$ large, when
$A<0$ and $\gamma\in(0,2)$. Such surfaces are close to sections of unduloids
with small necksize, folded along circumferences centered at the origin and
with larger and larger radii. The construction involves a deep study of the
corresponding Jacobi operators, an application of the Lyapunov-Schmidt
reduction method and some variational argument.
