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 PDF

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

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 $\mathbb {C}^2$.

References

Zbigniew Błocki, Cauchy-Riemann meet Monge-Ampère, Bull. Math. Sci. 4 (2014), no. 3, 433–480. MR 3277882, DOI 10.1007/s13373-014-0058-2
Zbigniew Blocki, A lower bound for the Bergman kernel and the Bourgain-Milman inequality, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 2116, Springer, Cham, 2014, pp. 53–63. MR 3364678, DOI 10.1007/978-3-319-09477-9_{4}
Zbigniew Błocki and Włodzimierz Zwonek, Estimates for the Bergman kernel and the multidimensional Suita conjecture, New York J. Math. 21 (2015), 151–161. MR 3318425
Zbigniew Błocki and Włodzimierz Zwonek, On the Suita conjecture for some convex ellipsoids in $\Bbb {C}^2$, Exp. Math. 25 (2016), no. 1, 8–16. MR 3424829, DOI 10.1080/10586458.2014.1002871
C. K. Cheung and Kang-Tae Kim, Analysis of the Wu metric. I. The case of convex Thullen domains, Trans. Amer. Math. Soc. 348 (1996), no. 4, 1429–1457. MR 1357392, DOI 10.1090/S0002-9947-96-01642-X
C. K. Cheung and K. T. Kim, Analysis of the Wu metric. II. The case of non-convex Thullen domains, Proc. Amer. Math. Soc. 125 (1997), no. 4, 1131–1142. MR 1363414, DOI 10.1090/S0002-9939-97-03695-2
John P. D’Angelo, A note on the Bergman kernel, Duke Math. J. 45 (1978), no. 2, 259–265. MR 473231
John P. D’Angelo, An explicit computation of the Bergman kernel function, J. Geom. Anal. 4 (1994), no. 1, 23–34. MR 1274136, DOI 10.1007/BF02921591
Ian Graham, Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in $C^{n}$ with smooth boundary, Trans. Amer. Math. Soc. 207 (1975), 219–240. MR 372252, DOI 10.1090/S0002-9947-1975-0372252-8
Lars Hörmander, $L^{2}$ estimates and existence theorems for the $\bar \partial$ operator, Acta Math. 113 (1965), 89–152. MR 179443, DOI 10.1007/BF02391775
Nikolai Nikolov, Localization of invariant metrics, Arch. Math. (Basel) 79 (2002), no. 1, 67–73. MR 1923040, DOI 10.1007/s00013-002-8286-1
Marek Jarnicki and Peter Pflug, Invariant distances and metrics in complex analysis, Second extended edition, De Gruyter Expositions in Mathematics, vol. 9, Walter de Gruyter GmbH & Co. KG, Berlin, 2013. MR 3114789, DOI 10.1515/9783110253863
László Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), no. 4, 427–474 (French, with English summary). MR 660145, DOI 10.24033/bsmf.1948
László Lempert, Holomorphic invariants, normal forms, and the moduli space of convex domains, Ann. of Math. (2) 128 (1988), no. 1, 43–78. MR 951507, DOI 10.2307/1971462
S. I. Pinčuk, Holomorphic inequivalence of certain classes of domains in $\textbf {C}^{n}$, Mat. Sb. (N.S.) 111(153) (1980), no. 1, 67–94, 159 (Russian). MR 560464
Harish Seshadri and Kaushal Verma, On isometries of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 5 (2006), no. 3, 393–417. MR 2274785