# Gradient estimate for harmonic functions on K\"ahler manifolds

## TL;DR

This paper establishes a precise gradient estimate for harmonic functions on noncompact K"ahler manifolds, leading to spectral bounds and a splitting theorem, advancing understanding of geometric analysis in complex manifolds.

## Contribution

It provides a sharp integral gradient estimate for harmonic functions on noncompact K"ahler manifolds, with applications to spectral theory and manifold splitting.

## Key findings

- Sharp gradient estimate for harmonic functions
- Bounds on the bottom of the spectrum of the p-Laplacian
- Splitting theorem for manifolds attaining the estimate

## Abstract

We prove a sharp integral gradient estimate for harmonic functions on noncompact K\"ahler manifolds. As application, we obtain a sharp estimate for the bottom of spectrum of the p-Laplacian and prove a splitting theorem for manifolds achieving this estimate.