BOUNDARY PRESERVATION BY ANALYTIC MAPS BETWEEN BORDERED RIEMANN SURFACES

摘要

This note shows that an analytic map from one compact bordered Riemann surface onto another must map the boundary of the domain surface onto the boundary of the image. To establish this result, we use a certain rigidity property of analytic arcs. In light of this discussion, we note that the Riemann-Hurwitz formula for an analytic map between bordered Riemann surfaces hold when the condition of boundary preservation is replaced by the condition that the map in question be surjective.