Teichmüller's uniqueness theorem ($K(f)\ge K(f_0),K(f)=K(f_0)\Rightarrow f=f_0$)

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.