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Remarks on the higher dimensional Suita conjecture


Authors: G. P. Balakumar, Diganta Borah, Prachi Mahajan and Kaushal Verma
Journal: Proc. Amer. Math. Soc. 147 (2019), 3401-3411
MSC (2010): Primary 32F45, 32A07, 32A25
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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Additional Information

G. P. Balakumar
Affiliation: Department of Mathematics, Indian Institute of Technology Palakkad, 678557, India
Email: gpbalakumar@gmail.com

Diganta Borah
Affiliation: Indian Institute of Science Education and Research, Pune 411008, India
Email: dborah@iiserpune.ac.in

Prachi Mahajan
Affiliation: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
Email: prachi@math.iitb.ac.in

Kaushal Verma
Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India
Email: kverma@iisc.ac.in

DOI: https://doi.org/10.1090/proc/14421
Keywords: Suita conjecture, Bergman kernel, Kobayashi indicatrix
Received by editor(s): August 29, 2018
Received by editor(s) in revised form: October 3, 2018
Published electronically: May 8, 2019
Additional Notes: The second-named author was partially supported by the DST-INSPIRE grant IFA-13 MA-21.
Communicated by: Harold P. Boas
Article copyright: © Copyright 2019 American Mathematical Society