Recovering $U(n)$-invariant toric K\"ahler metrics on $\mathbb{C}\mathbb{P}^n$ from the torus equivariant spectrum

TL;DR

This paper proves that $U(n)$-invariant toric K"ahler metrics on complex projective space are uniquely determined by their equivariant spectrum, combining spectral data with symmetry considerations.

Contribution

It establishes that the equivariant spectrum completely determines $U(n)$-invariant toric K"ahler metrics on $bC P^n$, a novel spectral rigidity result.

Findings

Equivariant spectrum uniquely determines the metric

Spectral data combined with symmetry fully characterizes the metrics

Results apply specifically to $U(n)$-invariant toric K"ahler metrics

Abstract

In this note we prove that toric K\"ahler metrics on complex projective space which are also $U(n)$-invariant are determined by their equivariant spectrum i.e. the list of eigenvalues of the Laplacian together with weights of the torus representation on the eigenspaces.