Extremal length of weak homology classes on Riemann surfaces

Abstract

1. The square integrable harmonic differentials on a Riemann surface W form a Hilbert space rh. Let rP be a closed subspace of rh. Let c be a 1-chain on W. There exists a unique element i1(c) CrP with the property flco= (c, {1'(c)) for all coCPr. We refer to it(c) as the F.reproducing differential for c. Accola [1] has shown that if c is a cycle, then the extremal length of the homology class of c is equal to the square of the norm of the rh-reproducing differential for c (cf. also [3]). Two specific problems raised by Accola's result are the following. For the important subspaces rF, does the norm of the rFreproducing differential for a cycle have an extremal length interpretation? Secondly, we may ask for a family of curves associated with a 1-chain c, not necessarily a cycle, whose extremal length gives the norm of the rh-reproducer for c. (By Abel's theorem, the vanishing of the norm of this reproducer implies that ac is a principal divisor.) In the present paper we give an answer to the first question for the subspace rh8e. Theorem 1 states that an associated geometric configuration is the weak homology class of c.2