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