Nevanlinna-type theorems for meromorphic functions on non-positively curved Kähler manifolds
The authors give a second main theorem of Nevanlinna theory for complete non-positively curved Kähler manifolds. Let \(M\) be a complete Kähler manifold of complex dimension \(n\) and let \(f\) be a nonconstant meromorphic function on \(M\). Analogous of the characteristic function and the proximity function of \(f\) are defined by \[ \begin{gathered} \tilde{T}_x(t)=\int\limits_0^t\int\limits_Mp(s,x,y)f^*\omega_{FS}\wedge\frac{\omega^{n-1}}{(n-1)!}\,ds, \\ \tilde{m}_x(t,a)=\int\limits_Mp(t,x,y)\log[f(y),a]^{-2}\,dV(y)\,, \end{gathered} \] where \(p(t,x,y)\) is the minimal fundamental solution of \(\frac{\partial u}{\partial t}-\frac12\Delta_Mu=0\) \(\Delta_M\) is the Laplacian defined by the Kähler metric, \(\omega_{FS}\) is the Fubini-Study metric on \(\mathbb{P}^1(\mathbb{C})\), \(\omega\) is the Kähler metric of \(M\) and \([w, a]\) is the chordal distance on \(\mathbb{P}^1(\mathbb{C})\). There exist \(\beta<2\) and some \(C >0\) such that the pointwise lower bound of the Ricci curvature \(R(x)\) satisfies \(R(x)\geq-Cr(x)^\beta-C\), where \(r(x)\) is the distance function from a reference point \(x_0\). Let \(f\) be a nonconstant meromorphic function on \(M\) and let \(a_1,\dots,a_q\) be distinct points in \(\mathbb{P}^1(\mathbb{C})\) such that \(f(x)\neq a_j\), for \(j =1,\dots,q\). It is proved: If there exist \(C, c>0\) and \(\gamma<2\) such that \(T_x(r)\leq Ce^{cr^\gamma}\) and \(\lim_{t\to\infty}\tilde{T}_x(t)=\infty\), then \[ \sum\limits_{j=1}^q\tilde{m}_x(t,a_j)+\tilde{N}_1(t,x)\leq2\tilde{T}_x(t)+\tilde{N}_x(t,\operatorname{Ric})+(5+\delta)\log\tilde{T}_x(t)+O(1), \] except for \(t\) in an exceptional set of finite length. The characteristic functions \(T_x\), \(\tilde{N}_1\) and \(\tilde{N}_x\) are determined.
scientific article; zbMATH DE number 6825363
scientific article