Theory of Abelian differentials and relative extremal length with applications to extremal slit mappings

Let R be an arbitrary open Riemann surface and R be its Kerekjarto-Stoilow compactification. Partition the ideal boundary _??_ of R into three disjoint sets ƒ¿ , ƒÀ and ƒÁ . Consider a harmonic differential dU on R with finite Dirichlet norm •adU•aR<•‡ satisfying the boundary conditions on ƒÀ and ƒÁ: U , an integral of dU, is constant on each component ƒÀ' of ƒÀ with the constant so chosen that •çƒÀ' dU*=0 , dU* being the conjugate differential of dU, and dU*=0 along ƒÁ. Let An be the space of such differentials dU. For such the space Ah we shall first derive some consequences which are reduced to the well known theory of differentials (e. g . see Nevanlinna [34], and Ahlfors and Sario [5]) if we take the whole ideal boundary J as ƒ¿. Let ƒ¶ be a subregion of R which is not relatively compact . A normalized function u on ƒ¶ is defined as a single-valued harmonic function which takes the boundary value 0 on ƒ¿ and has the same boundary behaviors as dU• ̧Ah on ƒÀ and ƒÁ . In • ̃1, we shall establish the maximum-minimum principles (THEOREMS 1.1 and 1.2) , Green's formula (LEMMA 1.4) and the homology theorem (COROLLARY 1.3), etc. Further we shall show that for a normalized function u on ƒ¶ and dU• ̧Ah (ƒ¶) the equation