= Teichmüller's uniqueness theorem
{c}
{title2=$K(f)\geq K(f_0),\quad K(f)=K(f_0)\Rightarrow f=f_0$}
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.
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