Mathematical Research Letters

Volume 24 (2017)

Number 1

Bounding the first invariant eigenvalue of toric Kähler manifolds

Pages: 67 – 81



Stuart J. Hall (Department of Mathematics, Statistics and Physics, University of Newcastle, Newcastle upon Tyne, United Kingdom)

Thomas Murphy (Department of Mathematics, California State University, Fullerton, Calif., U.S.A.)


We generalise a theorem of Engman and Abreu–Freitas on the first invariant eigenvalue of non-negatively curved $S^1$-invariant metrics on $\mathbb{CP}^1$ to general toric Kähler metrics with non-negative scalar curvature. In particular, a simple upper bound of the first nonzero invariant eigenvalue for such metrics on complex projective space $\mathbb{CP}^n$ is exhibited. We derive an analogous bound in the case when the metric is extremal and a detailed study is made of the accuracy of the bound in the case of Calabi’s extremal metrics on $\mathbb{CP}^2 \sharp - \mathbb{CP}^2$.

Received 19 May 2015

Accepted 17 September 2015

Published 7 June 2017