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

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.