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Boundary behaviour of the span metric and its higher-order curvatures. (English) Zbl 1504.30057

Summary: In this note, we use scaling principle to study the boundary behaviour of the span metric and its higher-order curvatures on finitely connected Jordan planar domains. A localization of this metric near boundary points of finitely connected Jordan domains is also obtained. Further, we obtain boundary sharp estimates for the span metric near \(C^2\)-smooth boundary points on such domains.

MSC:

30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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[1] Ahlfors, L.; Beurling, A., Conformal invariants and function-theoretic null-sets, Acta Math., 83, 101-129 (1950) · Zbl 0041.20301
[2] Bergman, S.: The kernel function and conformal mapping, revised, American Mathematical Society, Providence, R.I., . Mathematical Surveys, No. V (1970) · Zbl 0208.34302
[3] Bergman, S.; Chalmers, B., A procedure for conformal mapping of triply-connected domains, Math. Comput., 21, 527-542 (1967) · Zbl 0154.41203
[4] Boas, HP, The Lu Qi-Keng conjecture fails generically, Proc. Am. Math. Soc., 124, 7, 2021-2027 (1996) · Zbl 0857.32010
[5] Borah, D.; Haridas, P.; Verma, K., Comments on the Green’s function of a planar domain, Anal. Math. Phys., 8, 3, 383-414 (2018) · Zbl 1400.30013
[6] Burbea, J., Capacities and spans on Riemann surfaces, Proc. Am. Math. Soc., 72, 2, 327-332 (1978) · Zbl 0419.30009
[7] Burbea, J., The higher order curvatures of weighted span metrics on Riemann surfaces, Arch. Math. (Basel), 43, 5, 473-479 (1984) · Zbl 0537.30010
[8] Conway, JB, Functions of one complex variable. II, Graduate Texts in Mathematics (1995), New York: Springer, New York · Zbl 0887.30003
[9] Falconer, KJ, The geometry of fractal sets, Cambridge tracts in mathematics (1986), Cambridge: Cambridge University Press, Cambridge
[10] Garabedian, PR; Schiffer, M., On existence theorems of potential theory and conformal mapping, Ann. Math., 2, 52, 164-187 (1950) · Zbl 0040.32903
[11] Greene, RE; Kim, KT; Krantz, SG, The geometry of complex domains, Progress in mathematics (2011), Boston: Birkhäuser Boston Inc, Boston · Zbl 1239.32011
[12] Sakai, M., Analytic functions with finite Dirichlet integrals on Riemann surfaces, Acta Math., 142, 3-4, 199-220 (1979) · Zbl 0406.30036
[13] Sakai, M., The sub-mean-value property of subharmonic functions and its application to the estimation of the Gaussian curvature of the span metric, Hiroshima Math. J., 9, 3, 555-593 (1979) · Zbl 0424.31002
[14] Sario, L., Nakai, M.: Classification theory of Riemann surfaces. In: Die Grundlehren der mathematischen Wissenschaften, Band 164. Springer, New York (1970) · Zbl 0199.40603
[15] Sarkar, AD; Verma, K., Boundary behaviour of some conformal invariants on planar domains, Comput. Methods Funct. Theory, 20, 1, 145-158 (2020) · Zbl 1442.30045
[16] Sugawa, T.: Unified approach to conformally invariant metrics on Riemann surfaces. In: Proceedings of the second ISAAC congress (Fukuoka, 1999), vol. 2, pp. 1117-1127. (2000) · Zbl 1075.30021
[17] Sugawa, T., A conformally invariant metric on Riemann surfaces associated with integrable holomorphic quadratic differentials, Math. Z., 266, 3, 645-664 (2010) · Zbl 1219.30020
[18] Suita, Nobuyuki, Capacities and kernels on Riemann surfaces, Arch. Ration. Mech. Anal., 46, 212-217 (1972) · Zbl 0245.30014
[19] Zarankiewicz, K., Uber ein numerisches verfahren zur konformen abbildung zweifach zusammenhängender Gebiete, Z. Angew. Math. Mech., 14, 2, 97-104 (1934) · Zbl 0009.02604
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