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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Capacities and spans on Riemann surfaces


Author: Jacob Burbea
Journal: Proc. Amer. Math. Soc. 72 (1978), 327-332
MSC: Primary 30C40; Secondary 30C75, 30C85, 30F10
DOI: https://doi.org/10.1090/S0002-9939-1978-0507333-7
MathSciNet review: 507333
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Abstract: Let $ K(z,z),R(z,z)$, and $ C(z)$ be the values of the Bergman kernel, the reduced Bergman kernel and the analytic capacity on an open Riemann surface $ \Omega $ (with respect to a local parameter z). Let $ M(z) = \pi K(z,z)$ and $ S(z) = \sqrt {\pi R(z,z)} $. For $ \Omega \notin {O_G}$ and for each integer $ n \geqslant 0$, it is shown that

$\displaystyle {C^{(n + 1)(n + 2)}} \leqslant (n + 1)!{\left( {\prod\limits_{k =... ...{k!} } \right)^{ - 2}}\det \left\Vert {{M_{j\bar k}}} \right\Vert _{j,k = 0}^n,$

where $ C = C(z)$ and $ {M_{j\bar k}} = ({\partial ^{j + k}}/\partial {z^j}\partial {\bar z^k})M(z)$. Equality occurs if and only if $ \Omega $ is conformally equivalent to the unit disk less (possibly) a closed set of inner capacity zero. The special case of this result, namely when $ n = 0$, is due to Hejhal and Suita. Let $ \kappa (z)$ be the curvature of the ``span metric'' $ S(z)\vert dz\vert$. As an attempt to resolve a conjecture of Suita, we also show that for $ \Omega \notin {O_{AD}},\kappa (z) \leqslant - 2$ for each $ z \in \Omega $. Both results are proved by studying suitable extremal problems.

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DOI: https://doi.org/10.1090/S0002-9939-1978-0507333-7
Keywords: Bergman kernel, analytic capacity, span, curvature
Article copyright: © Copyright 1978 American Mathematical Society