Abstract
We will show, among other things, that an entire holomorphic map \((f(z_1, z_2), g(z_1, z_2))\) from \(\mathbb {C}^2\) into the simple-looking surface \(x^2+y^2=1\) in \(\mathbb {C}^2\) reduces to constant if and only if \(f^{-1}_{z_2}(0)\subseteq g^{-1}_{z_1}(0)\) (ignoring multiplicities) unless \(g^{-1}_{z_1}(0)={\mathbb {C}}^2\). Applications to nonlinear partial differential equations and certain variations will also be given.
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Communicated by Irene Sabadini.
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This article is part of the topical collection “In memory of Carlos A. Berenstein (1944-2019)” edited by Irene Sabadini and Daniele Struppa.
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Li, B.Q., Lü, F. On Entire Solutions of a Pythagorean Functional Equation and Associated PDEs. Complex Anal. Oper. Theory 14, 50 (2020). https://doi.org/10.1007/s11785-020-01007-0
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DOI: https://doi.org/10.1007/s11785-020-01007-0