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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
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.
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