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A counterexample to a conjecture by Błocki-Zwonek. (English) Zbl 1394.32029

Let \(\Omega\subset\mathbb{C}^n\) be a bounded pseudoconvex domain. A conjecture of Błocki and Zwonek is that the function \(\beta(t):=\log\lambda(\{z\in\Omega: g_{\Omega}(z,a) < t \})\) is convex, where \(g_{\Omega}(z,a)\) denotes the pluricomplex Green function for \(\Omega\) with logarithmic pole at \(a\in\Omega\) and \(\lambda\) denotes the Lebesgue measure in \(\mathbb{C}^n\).
The authors give a one-variable counterexample to this conjecture, where \(\Omega\) is the annulus \(\{z:\frac{1}{2}<|z|<2\}\) and \(a=1\). They show numerically that \(\beta(t)\) becomes concave when \(t\) exceeds the critical value \(t_c\); the set \(\{g_{\Omega}(z,a) < t\}\) is simply connected when \(t < t_c\) and homeomorphic to the annulus when \(t>t_c\).

MSC:

32U35 Plurisubharmonic extremal functions, pluricomplex Green functions
32T99 Pseudoconvex domains
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[1] Åhag, [Åhag And Czyż 15A] P.; Czyż, R., On the Błocki-Zwonek Conjectures, Complex Var. Elliptic Equ., 60, 9, 1270-1276 (2015) · Zbl 1343.32028
[2] Åhag, [Åhag And Czyż 15B] P.; Czyż, R., On the Błocki-Zwonek Conjectures and Beyond, Arch. Math. (Basel), 105, 4, 371-380 (2015) · Zbl 1326.32052
[3] Avelin, [Avelin 10A] H., Numerical Computations of Green’s Function and Its Fourier Coefficients on PSL(2,Z), Exp. Math., 19, 3, 335-343 (2010) · Zbl 1267.11062
[4] Avelin, [Avelin 10B] H., Computations of Green’s Function and Its Fourier Coefficients on Fuchsian Groups, Exp. Math., 19, 3, 317-334 (2010) · Zbl 1267.11061
[5] Berndtsson, [Berndtsson 98] B., Prekopa’s Theorem and Kiselman’s Minimum Principle for Plurisubharmonic Functions, Math. Ann., 312, 4, 785-792 (1998) · Zbl 0938.32021
[6] Berndtsson, [Berndtsson 06] B., Subharmonicity Properties of the Bergman Kernel and Some Other Functions Associated to Pseudoconvex Domains, Ann. Inst. Fourier (Grenoble), 56, 6, 1633-1662 (2006) · Zbl 1120.32021
[7] Błocki, [Błocki 14] Z.; Klartag, B.; Milman, E., Geometric Aspects of Functional Analysis, Israel Seminar (GAFA) 2011-2013, A Lower Bound for the Bergman Kernel and the Bourgain-Milman Inequality, 53-63 (2014), Cham: Springer, Cham
[8] Błocki, [Błocki 14] Z., Cauchy-Riemann Meet Monge-Ampère., Bull. Math. Sci., 4, 433-480 (2014) · Zbl 1310.32004
[9] Błocki, [Błocki And Zwonek 15] Z.; Zwonek, W., Estimates for the Bergman Kernel and the Multidimensional Suita Conjecture., New York J. Math., 21, 151-161 (2015) · Zbl 1317.32024
[10] Błocki, [Błocki And Zwonek 16] Z.; Zwonek, W., On the Suita Conjecture for Some Convex Ellipsoids in \(####\), Exp. Math., 25, 1, 8-16 (2016) · Zbl 1341.32010
[11] Fornæss, [Fornæss] J. E., On a Conjecture by Błocki-Zwonek · Zbl 1380.32034
[12] Green, G., An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism · ERAM 047.1264cj
[13] Green, [Green 50] G., An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism., J. Reine Angew. Math., 39, 73-89 (1850) · ERAM 039.1064cj
[14] Green, [Green 52] G., An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism., J. Reine Angew. Math., 44, 356-374 (1852) · ERAM 044.1211cj
[15] Green, [Green 54] G., An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism., J. Reine Angew. Math., 47, 161-221 (1854) · ERAM 047.1264cj
[16] Jarnicki, [Jarnicki And Pflug 13] M.; Pflug, P., Invariant Distances and Metrics in Complex Analysis, Second extended edition. de Gruyter Expositions in Mathematics, Vol., 9 (2013), Berlin: Walter de Gruyter GmbH & Co. KG, Berlin · Zbl 1273.32002
[17] Klimek, [Klimek 91] M., Pluripotential Theory, London Mathematical Society Monographs. New Series,, 6 (1991), New York: The Clarendon Press, Oxford University Press, New York · Zbl 0742.31001
[18] Krantz, [Krantz 13] S. G., Geometric Analysis of the Bergman Kernel and Metric, Graduate Texts in Mathematics,, 268 (2013), New York: Springer, New York · Zbl 1281.32004
[19] Wikström, [Wikström 03] F., Computing the Pluricomplex Green Function with Two Poles, Exp. Math., 12, 3, 375-384 (2003) · Zbl 1078.32021
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