Negatively curved K\"ahler metrics on total spaces of a class of vector bundles

TL;DR

This paper constructs numerous complete negatively curved K"ahler metrics on vector bundle total spaces, providing new insights into their geometric properties and holomorphic function behavior.

Contribution

It introduces a broad class of negatively curved K"ahler metrics on vector bundle total spaces and establishes dimension estimates and Liouville theorems for holomorphic functions and mappings.

Findings

Existence of complete negatively curved K"ahler metrics on certain vector bundle total spaces

Dimension bounds for holomorphic functions on these manifolds

Liouville theorems for holomorphic mappings between such manifolds

Abstract

In this paper we show an abundance of complete K\"ahler metrics with negative holomorphic bisectional curvature on total spaces of certain vector bundles. Assume that such total spaces are endowed with a wider class of nonpositively curved K\"ahler metrics. We prove dimension estimates on holomorphic functions on these manifolds, as well as Liouville theorems for holomorphic mappings between them.