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

Abstract

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 : R 3 Ñ R such that HpXq Ñ 1 as |X| Ñ 8 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.