Rie SASAI

*Generalized Obstacle Problem*

### Abstract

Fehlmann and Gardiner considered the obstacle problem
which asks what embedding of a Riemann surface *S* of finite topological type
minus an obstacle *E* into another surface *R* of the same type which
induces the isomorphisms π_{1}(*S*)->π_{1}(*R*) of the fundamental groups
does maximize the L^{1}-norm of the holomorphic quadratic differential on *R*
corresponding to a given one on *S* under the heights mapping.
In this paper we consider obstacles with arbitrarily many connected components
while they considered the case where the obstacle has finitely many
components.
As an application we give a slit mapping theorem of an open
Riemann surface of finite genus.

submission: 24 February 2004

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