It is considered the class of Riemann surfaces with dimT1 = 0, where T1 is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the space ΩH of harmonic forms of the surface, namelyΩH = *Ω0H +T1 + *T0H + T0H + T2.The surfaces in the class OHD and the class of planar surfaces satisfy dimT1 = 0. A.Pfluger posed the question whether there might exist other surfaces outside those two classes. Here it is shown that in the case of finite genus g, we should look for a surface S with dimT1 = 0 among the surfaces of the form SgK , where Sg is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary.