Capacities and spans on Riemann surfaces

Source abstract

Let K ( z , z ) , R ( z , z ) K(z,z),R(z,z) , and C ( z ) C(z) be the values of the Bergman kernel, the reduced Bergman kernel and the analytic capacity on an open Riemann surface Ω \Omega (with respect to a local parameter z ). Let M ( z ) = π K ( z , z ) M(z) = \pi K(z,z) and S ( z ) = π R ( z , z ) S(z) = \sqrt {\pi R(z,z)} . For Ω ∉ O G \Omega \notin {O_G} and for each integer n ⩾ 0 n \geqslant 0 , it is shown that $$C ( n + 1 ) ( n + 2 ) ⩽ ( n + 1 ) ! ( ∏ k = 0 n + 1 k ! ) − 2 det ‖ M j k ¯ ‖ j , k = 0 n , {C^{(n + 1)(n + 2)}} \leqslant (n + 1)!{\left ( {\prod \limits _{k = 0}^{n + 1} {k!} } \right )^{ - 2}}\det \left \| {{M_{j\bar k}}} \right \|_{j,k = 0}^n,$$ where C = C ( z ) C = C(z) and M j k ¯ = ( ∂ j + k / ∂ z j ∂ z ¯ k ) M ( z ) {M_{j\bar k}} = ({\partial ^{j + k}}/\partial {z^j}\partial {\bar z^k})M(z) . Equality occurs if and only if Ω \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 n = 0 , is due to Hejhal and Suita. Let κ ( z ) \kappa (z) be the curvature of the “span metric” S ( z ) | d z | S(z)|dz| . As an attempt to resolve a conjecture of Suita, we also show that for Ω ∉ O A D , κ ( z ) ⩽ − 2 \Omega \notin {O_{AD}},\kappa (z) \leqslant - 2 for each z ∈ Ω z \in \Omega . Both results are proved by studying suitable extremal problems.