Capacities and spans on Riemann surfaces

Burbea, Jacob

doi:10.1090/S0002-9939-1978-0507333-7

Capacities and spans on Riemann surfaces
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by Jacob Burbea PDF

Proc. Amer. Math. Soc. 72 (1978), 327-332 Request permission

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 \[ {C^{(n + 1)(n + 2)}} \leqslant (n + 1)!{\left ( {\prod \limits _{k = 0}^{n + 1} {k!} } \right )^{ - 2}}\det \left \| {{M_{j\bar k}}} \right \|_{j,k = 0}^n,\] 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” $S(z)|dz|$. 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.

References

Dennis A. Hejhal, Theta functions, kernel functions, and Abelian integrals, Memoirs of the American Mathematical Society, No. 129, American Mathematical Society, Providence, R.I., 1972. MR 0372187
Dennis A. Hejhal, Some remarks on kernel functions and Abelian differentials, Arch. Rational Mech. Anal. 52 (1973), 199–204. MR 344453, DOI 10.1007/BF00247732
L. Sario and K. Oikawa, Capacity functions, Die Grundlehren der mathematischen Wissenschaften, Band 149, Springer-Verlag New York, Inc., New York, 1969. MR 0254232
Menahem Schiffer, The span of multiply connected domains, Duke Math. J. 10 (1943), 209–216. MR 8259
Nobuyuki Suita, Capacities and kernels on Riemann surfaces, Arch. Rational Mech. Anal. 46 (1972), 212–217. MR 367181, DOI 10.1007/BF00252460

Über ein numerisches Verfahren zur konformen Abbildung zweifach zusammenhängender Gebiete

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