First eigenvalue of the $p$-Laplacian on K\"ahler manifolds

TL;DR

This paper establishes a lower bound for the first eigenvalue of the p-Laplacian on K"ahler manifolds, extending classical results and improving bounds by leveraging Hessian decomposition in positively curved cases.

Contribution

It provides a Lichnerowicz type lower bound for the p-Laplacian eigenvalue on K"ahler manifolds, generalizing known results for p=2 and improving bounds with Hessian decomposition techniques.

Findings

Derived a lower bound for the first eigenvalue of the p-Laplacian.

Extended classical eigenvalue bounds to K"ahler manifolds.

Utilized Hessian decomposition to improve bounds in positive Ricci curvature cases.

Abstract

We prove a Lichnerowicz type lower bound for the first nontrivial eigenvalue of the $p$-Laplacian on K\"ahler manifolds. Parallel to the $p = 2$ case, the first eigenvalue lower bound is improved by using a decomposition of the Hessian on K\"ahler manifolds with positive Ricci curvature.