Communications in Analysis and Geometry

Volume 31 (2023)

Number 9

Kähler–Einstein metrics and eigenvalue gaps

Pages: 2255 – 2275

DOI: https://dx.doi.org/10.4310/CAG.2023.v31.n9.a4

Authors

Bin Guo (Department of Mathematics and Computer Sciences, Rutgers University, Newark, New Jersey, U.S.A.)

Duong H. Phong (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)

Jacob Sturm (Department of Mathematics and Computer Sciences, Rutgers University, Newark, New Jersey, U.S.A.)

Abstract

The existence of Kähler–Einstein metrics on a Fano manifold is characterized in terms of a uniform gap between $0$ and the first positive eigenvalue of the Cauchy–Riemann operator on smooth vector fields. It is also characterized by a similar gap between $0$ and the first positive eigenvalue for Hamiltonian vector fields. The underlying tool is a compactness criteria for suitably bounded subsets of the space of Kähler potentials which implies a positive gap.

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The authors’ work was supported in part by the National Science Foundation under grants DMS-1855947 and DMS-1945869.

Received 20 January 2021

Accepted 27 October 2021

Published 12 August 2024