 Teichmüller's uniqueness theorem

On a closed genus-at-least-two Riemann surface, a Teichmüller map uniquely minimizes maximal dilatation in its homotopy class among maps to the same target. The Reich–Strebel inequality forces equality of Beltrami coefficients for an extremal competitor; their conformal difference is homotopic to the identity and hence is the identity. In genus one, translations must be factored out.