On a class of Riemann surfaces

Abstract

It is considered the class of Riemann surfaces with dim T ₁ = 0, where T ₁ is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the space Θ^H of harmonic forms of the surface, namely

$$\Omega ^H = *\Omega _0^{\rm H} \dot + T_1 \dot + *T_0^H \dot + T_0^H \dot + T_2 $$

.

The surfaces in the class O_HD and the class of planar surfaces satisfy dim T ₁ = 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 dim T ₁ = 0 among the surfaces of the form S_g/K, where S_g is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary.