Capacity Functions

On analytic self-mappings of Riemann surfaces

“…16 will largely follow them, more precise arguments will be needed. It is known that 2(F(c)) is equal to the harmonic length h(c) of c [12]. It seems to us that it has less connection with ours.…”

Conformal Embeddings of an Open Riemann Surface into Another — A Counterpart of Univalent Function Theory —

“…Many proofs are known today. Some of the typical proofs can be found, for example, in [5], [8], [14], [19], and [29]. Similarly, there uniquely exists an element f 1 of F which minimizes Re f .…”

“…(For other topics, see for instance, [38], [23], [39], and [40].) As an example of such topics we focus our attention to the span in the second half of the paper.The span is first defined by M. Schiffer ([30]) for multiply connected plain domains, and has been since then generalized in many different directions (see e.g., [15], [27], [28], [29]). Hamano-Maitani-Yamaguchi [10] considered a certain holomorphic family of finite Riemann surfaces parametrized by the unit disk D and showed that one of the generalized spans (the harmonic span in [28]) is a subharmonic function on D. Concerning the Schiffer span for plane domains a similar result is obtained by S. Hamano ([9]).…”

“…If there is an injective holomorphic mapping of R into another Riemann surfaceR, then the pair ðR;Þ is called a continuation (or prolongation, extension) of R. (cf. [29]). Specifically, ðR;Þ is called a compact continuation of R ifR is a closed surface, and a continuation of the same genus if the genus ofR is the same as that of R. In this article we use a shorter term ''closing'' instead of the long phrase ''compact continuation of the same genus.'…”

“…Cf. [5], [8], and [29]. Z. Nehari [25] and M. Mori [24] intended to generalize this classical theorem to open Riemann surfaces of finite genus.…”