Bounding $\lambda_{2}$ for K\"ahler-Einstein metrics with large symmetry groups

TL;DR

This paper derives an upper bound for the second eigenvalue of the scalar Laplacian on toric K"ahler-Einstein manifolds using polytope data, with detailed examples and potential extensions.

Contribution

It introduces a method to bound $$ eigenvalues for toric K"ahler-Einstein metrics based on polytope data, expanding understanding of spectral geometry in this context.

Findings

Derived explicit upper bounds for $$ eigenvalues

Provided detailed examples in complex dimensions 1, 2, and 3

Discussed potential extensions to other geometries

Abstract

We calculate an upper bound for the second nonzero eigenvalue of the scalar Laplacian, $\lambda_{2}$, for toric K\"ahler-Einstein metrics in terms of the polytope data. We give some detailed examples in complex dimensions 1, 2 and 3. We also discuss extensions of this method to other geometries.