Spectrum of the Laplacian on the Page metric

TL;DR

This paper numerically computes the Laplacian spectrum on the Page Einstein metric of a complex projective plane connected sum, using Sturm-Liouville reduction, perturbative analysis, and pseudospectral methods.

Contribution

It introduces a numerical approach to determine the Laplacian spectrum on the Page metric, combining Sturm-Liouville reduction with perturbative and pseudospectral techniques.

Findings

Spectrum computed numerically for the Page metric

Perturbative analysis supports the numerical results

Spectrum of the Lichnerowicz Laplacian also studied

Abstract

We numerically construct the spectrum of the Laplacian on Page's inhomogeneous Einstein metric on $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$ by reducing the problem to a (singular) Sturm-Liouville problem in one dimension. We perform a perturbative analysis based upon a closely related, exactly solvable problem that strongly supports our results. We also study the spectrum of the Lichnerowicz Laplacian on symmetric traceless transverse two-tensors. The method relies on both the isometries of the Page metric and pseudospectral methods to numerically solve the resulting ODEs.