Remarks on the higher dimensional Suita conjecture

Balakumar, G. P.; Borah, Diganta; Mahajan, Prachi; Verma, Kaushal

doi:10.1090/proc/14421

Remarks on the higher dimensional Suita conjecture
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by G. P. Balakumar, Diganta Borah, Prachi Mahajan and Kaushal Verma

Proc. Amer. Math. Soc. 147 (2019), 3401-3411

DOI: https://doi.org/10.1090/proc/14421

Published electronically: May 8, 2019

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Abstract:

To study the analog of Suita’s conjecture for domains $D \subset \mathbb {C}^n$, $n \geq 2$, Błocki introduced the invariant $F^k_D(z)=K_D(z)\lambda \big (I^k_D(z)\big )$, where $K_D(z)$ is the Bergman kernel of $D$ along the diagonal and $\lambda \big (I^k_D(z)\big )$ is the Lebesgue measure of the Kobayashi indicatrix at the point $z$. In this note, we study the behaviour of $F^k_D(z)$ (and other similar invariants using different metrics) on strongly pseudconvex domains and also compute its limiting behaviour explicitly at certain points of decoupled egg domains in $\mathbb {C}^2$.

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