Embedded tori with prescribed mean curvature

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|−γ for |X| large, when A < 0 and γ ∈ (0,2). Such surfaces are close to sections of unduloids with small neck-size, 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.