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Abstract
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We consider the Riemann surface X obtained by removing n points from the Riemann sphere. It is well known that X has the hyperbolic metric when n is at least three. In this talk, we consider mainly the case when n>3. Our final goal is to construct an explicitly expressed distance function on X which is comparable with the hyperbolic distance on X. To this end, we propose a quantity given in terms of the cross ratios of four points from the punctures to detect locations of collars as well as its width. This is joint work with Tanran Zhang at Soochow University and, partly, with Matti Vuorinen at University of Turku.
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