Yamada's extended Suita conjecture

From papers

Let $\Omega$ be an open Riemann surface admitting a nontrivial Green function $G_\Omega$ , let $u$ be harmonic on $\Omega$ , and set $\rho=e^{-2u}$ . Define $B_{\Omega,\rho}$ from an orthonormal basis of weighted $L^2$ holomorphic $1$ -forms satisfying the completeness condition stated in the source. Let $c_\beta(z_0)$ be the logarithmic capacity associated with $G_\Omega(\cdot,z_0)$ , and let $\chi_{-u}$ and $\chi_{z_0}$ be the characters associated with $-u$ and $G_\Omega(\cdot,z_0)$ , respectively. Yamada's extended Suita conjecture.

c_\beta(z_0)^2\le\pi\rho(z_0)B_{\Omega,\rho}(z_0)

for any $z_0\in\Omega$ , and equality holds if and only if $\chi_{-u}=\chi_{z_0}$ . The inequality part of the extended Suita conjecture was proved by Guan and Zhou; the equality characterization is included in the resolved statement described by the source.

Progress summary

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Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Yamada's extended Suita conjecture
Let $\Omega$ be an open Riemann surface with Green function, let $z_0\in\Omega$ , and let $u$ be a harmonic function on $\Omega$ . Write $B_{\Omega,e^{-2u}}(z_0)$ for the weighted Bergman kernel and let $\chi_{z_0}$ and $\chi_{-u}$ be the characters associated with the Green function $G_{\Omega}(\cdot,z_0)$ and the harmonic function $-u$ , respectively. Yamada's extended Suita conjecture.
$c_{\beta}^2(z_0)\leq \pi e^{-2u(z_0)}B_{\Omega,e^{-2u}}(z_0)$
and equality holds if and only if $\chi_{z_0}=\chi_{-u}$ . The source introduces this as a conjecture, but the supplied text does not establish its resolution; its status is therefore left open.
source: Qi'an Guan, Xun Sun and Zheng Yuan, “A remark on a weighted version of Suita conjecture for higher derivatives”, arXiv:2212.02713 (2022).

Sources & referencesView supporting material

Primary source

Qi'an Guan and Zheng Yuan, “Concavity property of minimal L^2 integrals with Lebesgue measurable gain IV: product of open Riemann surfaces”, arXiv:2211.04953 (2022).

Additional references

3 papers in this index state this conjecture (2022). The statement above is taken from the most recent of them; the others are arXiv:2211.00470, arXiv:2207.10976.

Source: https://arxiv.org/abs/2211.04953 Yamada (year not specified), cited in the source as Yamada Guan-Zhou (2015), cited in the source as GZ15

1 week ago