Recovering $U(n)$-invariant toric K\"ahler metrics on   $\mathbb{C}\mathbb{P}^n$ from the torus equivariant spectrum
  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. 
Tom\'as A. Reis and Rosa Sena-Dias
math.DG math.SP