A boundary value problem in the hyperbolic space.
Simple computations support the conjecture that a small spherical surface with its center on a minimal surface cannot be divided by the minimal surface into two portions with different area.
We study the problem of existence of surfaces in R3 parametrized on the sphere S2 with prescribed mean curvature H in the perturbative case, i.e. for H = Ho + EH1, where Ho is a nonzero constant, H1 is a C2 function and E is a small perturbation parameter.