Ahlfors extremals realizing the gonality

Let a compact bordered Riemann surface have gonality equal to the least degree of a conformal map to the disc, and let fa,bf_{a,b} denote an Ahlfors extremal function associated with two marked points, with faf_a the limiting one-point extremal. Ahlfors gonality conjecture. Any conformal mapping realizing the gonality arises as an Ahlfors extremal function fa,bf_{a,b}, perhaps after coalescing two points to obtain an Ahlfors map faf_a maximizing the modulus of the derivative at aa. The source records the question as unsettled; it is proved there only in the hyperelliptic case.

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Primary source

Alexandre Gabard, “Notes on Hilbert's 16th: experiencing Viro's theory”, arXiv:1310.1865 (2013).

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