Notes on theta functions for open Riemann surfaces (Q795961)

Let W be an arbitrary open Riemann surface of positive (possibly infinite) genus. Using the notion of normal behaviour spaces of harmonic differentials introduced by \textit{F. Maitani} [J. Math. Kyoto Univ. 20, 661-689 (1980; Zbl 0494.30039)] the author introduces and studies a certain subspace of the Hilbert space of square integrable complex differentials on W. In terms of this space and its dual, he obtains a candidate for a Jacobi variety for W, and determines an analogue of the Jacobi mapping (reducing to the usual objects for compact W). He furthermore obtains analogues of the theta-functions and studies various properties of these. Relations to the work of \textit{H. P. McKean} and \textit{E. Trubowitz} [Bull. Am. Math. Soc. 84, 1042-1085 (1978; Zbl 0428.34026)] are considered.