Extremal Kähler Metrics Induced by Finite or Infinite-Dimensional Complex Space Forms

In this paper, we address the problem of studying those complex manifolds M equipped with extremal metrics g induced by finite or infinite-dimensional complex space forms. We prove that when g is assumed to be radial and the ambient space is finite-dimensional, then ( M ,  g ) is itself a complex sp...

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Bibliographic Details
Published in:The Journal of geometric analysis Vol. 31; no. 8; pp. 7842 - 7865
Main Authors: Loi, Andrea, Salis, Filippo, Zuddas, Fabio
Format: Journal Article
Language:English
Published: New York Springer US 01.08.2021
Springer Nature B.V
Subjects:
Abstract Harmonic Analysis
Convex and Discrete Geometry
Curvature
Differential Geometry
Dynamical Systems and Ergodic Theory
Fourier Analysis
Geometry
Global Analysis and Analysis on Manifolds
Mathematics
Mathematics and Statistics
extremal metric
32Q15
Complex space forms
53C55
Constant scalar curvature metric
Kählermetric
Calabi’s diastasis function
32T15
ISSN:1050-6926, 1559-002X
Online Access:Get full text
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Summary:In this paper, we address the problem of studying those complex manifolds M equipped with extremal metrics g induced by finite or infinite-dimensional complex space forms. We prove that when g is assumed to be radial and the ambient space is finite-dimensional, then ( M ,  g ) is itself a complex space form. We extend this result to the infinite-dimensional setting by imposing the strongest assumption that the metric g has constant scalar curvature and is well behaved (see Definition 1 in the Introduction). Finally, we analyze the radial Kähler–Einstein metrics induced by infinite-dimensional elliptic complex space forms and we show that if such a metric is assumed to satisfy a stability condition then it is forced to have constant nonpositive holomorphic sectional curvature.
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ISSN:1050-6926
1559-002X
DOI:10.1007/s12220-020-00554-4