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In Euclidean 3-space endowed with a Cartesian reference system we consider aclass of surfaces, called Delaunay tori, constructed by bending segments ofDelaunay cylinders with neck-size $a$ and $n$ lobes along circumferencescentered at the origin. Such surfaces are complete and compact, have genus oneand almost constant, say 1, mean curvature, when $n$ is large. Considering aclass 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 asnormal graph over the Delaunay torus and whose mean curvature equals $H$ atevery point.

On the non-existence of compact surfaces of genus one with prescribed, almost constant mean curvature, close to the singular limit