The complexity of a branched cover of a Riemann surface M to the Riemann sphere S2 is defined as its degree times the hyperbolic area of the complement of its branching set in S2. The complexity of M is defined as the infimum of the complexities of all branched covers of M to S2. We prove that if M is a connected, closed, orientable Riemann surface of genus g≥1, then its complexity equals 2π(mmin+2g−2) , where mmin is the minimum total length of a branch datum realisable by a branched cover p:M→S2.
