The moduli of compact continuations of an open Riemann surface of genus one

Let $(R, \{ A, B\} )$ be a marked open Riemann surface of genus one. Denote by $(T, \{ {A_T}, {B_T}\} ,i)$ a pair of a marked torus $(T, \{ {A_T}, {B_T}\} )$ and a conformal embedding $i$ of $R$ into $T$ with $i(A)$ and $i(B)$ homotopic respectively to ${A_T}$ and ${B_T}$. We say that $(T, \{ {A_T}, {B_T}\} ,i)$ and $(T’, \{ {A_T’}, {B_T’}\} ,i’)$ are equivalent if $i’ \circ {i^{ - 1}}$ extends to a conformal mapping of $T$ onto ${T^\prime }$. The equivalence classes are called compact continuations of $(R, \{ A, B\} )$ and the set of moduli of compact continuations of $(R, \{ A, B\} )$ is denoted by $M = M(R, \{ A, B\} )$. Then $M$ is a closed disk in the upper half plane. The radius of $M$ represents the size of the ideal boundary of $R$ and gives a generalization of Schiffer’s span for planar domains; in particular, it vanishes if and only if $R$ belongs to the class ${O_{AD}}$. On the other hand, any holomorphic differential on $R$ with distinguished imaginary part produces in a canonical manner a compact continuation of $(R, \{ A, B\} )$. Such a compact continuation is referred to as a hydrodynamic continuation of $(R, \{ A, B\} )$. The boundary of $M$ parametrizes in a natural way the space of hydrodynamic continuations; i.e., the hydrodynamic continuations have extremal properties.