Two remarks on Riemann surfaces
Let \(M\) be a complete Riemannian manifold of dimension 2 with a metric of constant curvature \(-1\). One says \(M\) satisfies a linear isoperimetric inequality (LII) if there exists a finite constant \(h(M)\) so that for every bounded open set \(G\) with smooth boundary, \(A(G) \leq h (M) L(\partial G)\). The author discusses connections between LII and some conformal invariants. Let \({\mathcal B}\) be the set of Riemann surfaces which do not satisfy LII, let \({\mathcal O}_G\) be the set of Riemann surfaces which do not have a Green's function, let \({\mathcal O}_{HP}\) be the set of Riemann surfaces which do not have a nonconstant positive harmonic function; \({\mathcal O}_G \subset {\mathcal O}_{HP}\). It is known \({\mathcal O}_G \subset {\mathcal B}\); the author proves that \({\mathcal B}\) is not contained in \({\mathcal O}_{HP}\), even with the hypothesis that the injectivity radius \(\iota (M)\) is positive: Theorem 1: There exists a Riemann surface \({\mathcal M}\) with LII, \(\iota (M) > 0\), and such that every positive harmonic function on \({\mathcal M}\) is constant. Let \(L_M (p,t)\) be the length of the boundary of the ball of radius \(t\) centered at \(t \in M\). The author also proves: Theorem 2: There exists a Riemann surface \({\mathcal N}\) such that \({\mathcal N}\) does not have a Green's function and so \(L_{\mathcal N} (p,t) \geq ce^{t/2}\) for \(t \geq t_0\) and every \(p \in {\mathcal N}\).
scientific article; zbMATH DE number 791195