Mathematics Colloquia and Seminars

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Inscribed circles play an important role in classical plane geometry. In this talk we apply them to the study of Riemann surfaces. We present an elementary construction of a canonical Euclidean metric on any hyperbolic surface that varies continuously over moduli space. The metric is piecewise flat in the sense that it is smooth in the complement of a graph. This gives a natural decomposition of any closed Riemann surface into Euclidean pieces. Our construction also gives a canonical framing up to sign at most points.