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For a nonconstant holomorphic map $f:X\to Y$ of degree $d$ between compact Riemann surfaces, Riemann-Hurwitz formula is $2g_X-2=d(2g_Y-2)+{\sum }_p(e_p-1)$. Triangulate $Y$ with branch values as vertices and lift the triangulation: faces and edges lift $d$ times, while the vertex deficit is exactly the ramification sum. Comparing Euler characteristics proves the formula.

On closed Riemann surfaces of genus at least two, if a Teichmüller map $f_0:X\to Y$ has unit-area source differential $q$ and dilatation $K_0$, any quasiconformal map $f:X\to Y$ in the same homotopy class satisfies $K_0\le {\int }_X|q||1+{\mu }_{f}q/|q|{|}^2/(1-|{\mu }_f{|}^2)$. The plus sign corresponds to a Teichmüller map with ${\mu }_{f_0}=k_0|q|/q$ and $K_0=(1+k_0)/(1-k_0)$. This fundamental inequality yields Teichmüller's uniqueness theorem by the pointwise triangle inequality and its equality case. See Gardiner and Hu, §5, equation (11).

The study of deformations and markings of Riemann surfaces, using quasiconformal maps, holomorphic quadratic differentials, extremal length and the mapping class group.