A characterization of the algebraic surfaces on which the classical Phragmén-Lindelöf theorem holds using branch curves (Q652918)

An algebraic variety \(V\) in \(\mathbb{C}^n\), \(n \geq 2\), has the strong Phargmén-Lindelöf property (SPL) if there is a positive constant \(A\) such that each plurisubharmonic function \(u\) on \(V\) which is bounded above by \(|z|+o(|z|)\) on \(V\) and by \(0\) on the real points of \(V\) is already bounded by \(A|\text{Im} z|\) on the whole variety \(V\). There are several problems on linear partial differential operators \(P(D)\) with constant coefficients for which the property (SPL) on the zero variety of the polynomial or of its principal part plays a role. We refer the reader for example to [\textit{R. Meise} and \textit{B. A. Taylor}, Result. Math. 36, No. 1--2, 121--148 (1999; Zbl 0941.32032)], or to the authors' paper [J. Reine Angew. Math. 588, 169--220 (2005; Zbl 1088.47036)]. In the main theorem of the article under review the authors characterize those algebraic surfaces \(V\) in \(\mathbb{C}^n\) which satisfy (SPL). A different characterization was obtained by them in [Math. Z. 253, No. 2, 387--417 (2006; Zbl 1096.31003)]. It was later used by them to prove that an algebraic surface \(V\) in \(\mathbb{C}^n\) satisfies (SPL) if and only if each of its limit varieties satisfies also (SPL). The characterization presented now uses the behaviour of the branch curves with respect to many projections in \(\mathbb{C}^n\) which are noncharacteristic for \(V\) at infinity. In the final section it is shown how to apply the main theorem, and some concrete examples are discussed.