# Integral inequalities for holomorphic maps and applications

## TL;DR

This paper establishes integral inequalities for holomorphic maps between complex manifolds and uses them to prove rigidity, degeneracy, and singularity results in complex geometry, especially related to the Kähler-Ricci flow.

## Contribution

It introduces new integral inequalities for holomorphic maps and applies them to derive geometric rigidity, degeneracy theorems, and analyze the Kähler-Ricci flow without curvature sign assumptions.

## Key findings

- Rigidity and degeneracy theorems for holomorphic maps.
- Characterization of equality cases in certain settings.
- Insights into the infinite-time singularity types of Kähler-Ricci flow.

## Abstract

We derive some integral inequalities for holomorphic maps between complex manifolds. As applications, some rigidity and degeneracy theorems for holomorphic maps without assuming any pointwise curvature signs for both the domain and target manifolds are proved, in which key roles are played by total integration of the function of the first eigenvalue of second Ricci curvature and an almost nonpositivity notion for holomorphic sectional curvature introduced in our previous work. We also apply these integral inequalities to discuss the infinite-time singularity type of the K\"ahler-Ricci flow. The equality case is characterized for some special settings.