Kahler-Einstein Metrics and Eigenvalue Gaps

TL;DR

This paper characterizes the existence of Kahler-Einstein metrics on Fano manifolds through eigenvalue gaps of specific operators, providing a new spectral criterion for their existence.

Contribution

It introduces a novel spectral characterization of Kahler-Einstein metrics using eigenvalue gaps of the Cauchy-Riemann operator and Hamiltonian vector fields.

Findings

Eigenvalue gaps characterize Kahler-Einstein metrics.

A compactness criterion for Kahler potentials is established.

Positive eigenvalue gaps imply existence of Kahler-Einstein metrics.

Abstract

The existence of Kahler-Einstein metrics on a Fano manifold is characterized in terms of a uniform gap between 0 and the first positive eigenvalue of the Cauchy-Riemann operator on smooth vector fields. It is also characterized by a similar gap between 0 and the first positive eigenvalue for Hamiltonian vector fields. The underlying tool is a compactness criterion for suitably bounded subsets of the space of Kahler potentials which implies a positive gap.