Extremal quasiconformal mappings and classes of divisors on Riemann surfaces (Q1820896)

Let S be a closed Riemann surface of genus \(g>0\) and denote by J(S) the Jacobian variety of S, defined as the set of equivalence classes of divisors of degree zero. For \(\alpha\in J(S)\) and N a natural number, denote by \(\alpha_ N\) the set of divisors of type N in \(\alpha\). Furthermore, set \(J_ N(S)=\{\alpha_ N:\alpha\in J(S)\}\). The author shows: Theorem 1: Let \(F_ N(\alpha,\beta)\) be a non-empty family of quasiconformal automorphisms of S, carrying a divisor from \(\alpha_ N\) to \(\beta_ N\). Then \(F_ N(\alpha,\beta)\) contains an extremal quasiconformal map \(f_ 0\) that minimizes K[f]. \(f_ 0\) is the Teichmüller map of S, and its Beltrami coefficient is \(k\phi\) (z)/\(| \phi (z)|\) where \(0\leq k<1\) and \(\phi\) (z) is a quadratic differential that is regular on S except in the points of the divisor associated with \(f_ 0\) at which \(\phi\) (z) has simple poles. Let \(\alpha\),\(\beta\in J(S)\). If \(F_ N(\alpha,\beta)\neq \emptyset\), set \(\rho_ N(\alpha,\beta)=\ell n K[f_ 0]\) where \(f_ 0\) is an extremal map in \(F_ N(\alpha,\beta)\) that is homotopic to identity; otherwise set \(\rho_ N(\alpha,\beta)=\infty\). \(\rho_ N\) is a metric. Theorem \(2: \rho_ N\) is finite in some neighborhood of \(\alpha\in J(S)\) if and only if \(\alpha_ N\) contains a divisor \(\sum \{P_ j-Q_ j:\) \(j=1,...,N\}\) such that there is no non-zero abelian differential whose divisor is \(\geq \sum \{P_ j+Q_ j:\) \(j=1,...,N\}.\) Theorem 3: For \(N\geq g\), the canonical map \(J_ N(S)\to J(S)\) is a continuous surjection.