An extremal eigenvalue problem in K\"ahler geometry

TL;DR

This paper investigates extremal properties of Laplace eigenvalues on K"ahler manifolds, introducing a notion of extremal metrics and analyzing conditions for extremality, especially for K"ahler-Einstein metrics with positive scalar curvature.

Contribution

It defines a new concept of mbda_k-extremal K"ahler metrics and provides necessary and sufficient conditions for extremality, focusing on mbda_1 and K"ahler-Einstein metrics.

Findings

Characterization of mbda_k-extremal K"ahler metrics

Conditions for extremality in terms of geometric properties

Analysis of mbda_1-extremal properties of K"ahler-Einstein metrics

Abstract

We study Laplace eigenvalues $\lambda_k$ on K\"ahler manifolds as functionals on the space of K\"ahler metrics with cohomologous K\"ahler forms. We introduce a natural notion of a $\lambda_k$-extremal K\"ahler metric and obtain necessary and sufficient conditions for it. A particular attention is paid to the $\lambda_1$-extremal properties of K\"ahler-Einstein metrics of positive scalar curvature on manifolds with non-trivial holomorphic vector fields.