A refined form of the second main theorem on complete non-positively curved Kähler manifolds
Xianjing Dong
TL;DR
This paper addresses refining the second main theorem in Nevanlinna theory for meromorphic mappings on complete Kähler manifolds with non-positive sectional curvature. By introducing a new calculus lemma, it achieves a sharper main error term $O(\sqrt{-\kappa(r)}\,r)$, improving upon Atsuji’s $O(-\kappa(r)r^2)$. The approach combines Nevanlinna characteristic and proximity functions with a Green function calculus and Jacobi-ODE estimates, yielding the key inequality $T_f(r,L)+T_f(r,K_X)+T(r,\mathscr R) \le \overline N_f(r,D) + O(\log^+ T_f(r,L) + \sqrt{-\kappa(r)} r + \delta\log r)$ outside a finite-measure set. A constant-curvature corollary recovers the classical second main theorem for the unit ball and leads to a defect relation under mild curvature assumptions, thereby extending the classical theory to a broader geometric setting with tighter error terms.
Abstract
How to devise a second main theorem with best error terms is a central problem in the study of Nevanlinna theory. However, it seems difficult to be done for a general non-positively curved Kähler manifold. Based on the work of A. Atsuji in Nevanlinna theory, we present a refined form of the second main theorem of meromorphic mappings on a general complete Kähler manifold with non-positive sectional curvature using a good estimate. This result improves the error terms in the second main theorem obtained by A. Atsuji in 2018.