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Extremal quasiconformal mappings and classes of divisors on Riemann surfaces. (English. Russian original) Zbl 0615.30022

Sib. Math. J. 25, 507-509 (1984); translation from Sib. Mat. Zh. 25, No. 3(145), 204-206 (1984).
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.
Reviewer: H.Röhrl

MSC:

30C75 Extremal problems for conformal and quasiconformal mappings, other methods
30C62 Quasiconformal mappings in the complex plane
30F10 Compact Riemann surfaces and uniformization
14H40 Jacobians, Prym varieties
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References:

[1] G. Springer, Introduction to the Theory of Riemann Surfaces [Russian translation], IL, Moscow (1960). · Zbl 0094.05302
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[4] R. Rakhmankulov and É. Kh. Yakubov, ?An extremal quasiconformal mapping of closed Riemann surfaces,? Dokl. Akad. Nauk SSSR,218, No. 4, 761-763 (1974).
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