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 curva- ture, when n is large. Considering a class of mappings $H : R^3 → R$ such that $H(X) → 1$ as $|X| → ∞$ 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.