Abstract: An infinite Riemann surface is closest to a compact Riemann surface if it does not support a Green’s function. This condition is equivalent to the ergodicity of the geodesic flow as well as the transience of the Brownian motion. We give another characterization of this class of Riemann surfaces using the space of holomorphic quadratic differentials and prove that the set of holomorphic quadratic differentials with single cylinders is dense among all holomorphic quadratic differentials. It is well known that the class of Riemann surfaces without Green’s function is invariant under quasiconformal mappings. A class of Riemann surfaces without non-constant harmonic functions with finite Dirichlet integral properly contains the class of surfaces without Green’s function. We prove that the larger class is also invariant under quasiconformal mappings. The classes of Riemann surfaces that satisfy the (strong) Liouville property are sandwiched between the above two classes, and it is known that the (strong) Liouville property is not a quasiconformal invariant.

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