Mathematical Research Letters

Volume 24 (2017)

Number 1

Bounding the first invariant eigenvalue of toric Kähler manifolds

Pages: 67 – 81

DOI: http://dx.doi.org/10.4310/MRL.2017.v24.n1.a4

Authors

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.)

Abstract

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$.

Full Text (PDF format)

Paper received on 19 May 2015.