A Uniqueness Theorem for Holomorphic Mappings in the Disk Sharing Totally Geodesic Hypersurfaces

TL;DR

This paper extends Cartan's Second Main Theorem to holomorphic mappings in the disk intersecting geodesic hypersurfaces in projective space, establishing a new uniqueness result for such mappings with many intersections.

Contribution

It generalizes the Second Main Theorem to nonlinear hypersurfaces and proves a uniqueness theorem for holomorphic mappings intersecting numerous totally geodesic hypersurfaces.

Findings

Established a Second Main Theorem for mappings intersecting geodesic hypersurfaces.

Proved a uniqueness theorem for mappings intersecting O(k^3) hypersurfaces.

Generalized classical results to nonlinear hypersurfaces in projective space.

Abstract

In this paper, we prove a Second Main Theorem for holomorphic mappings in a disk whose image intersects some families of nonlinear hypersurfaces (totally geodesic hypersurfaces with respect to a meromorphic connection) in the complex projective space $\mathbb{P}^k$. This is a generalization of Cartan's Second Main Theorem. As a consequence, we establish a uniqueness theorem for holomorphic mappings which intersects $O(k^3)$ many totally geodesic hypersurfaces.