Abstract
Inspired by the work of Lu and Tian (Duke Math J 125(2):351–387, 2004), in this paper we address the problem of studying those Kähler manifolds satisfying the \(\Delta\)-property, i.e. such that on a neighborhood of each of its points the kth power of the Kähler Laplacian is a polynomial function of the complex Euclidean Laplacian, for all positive integer k (see below for its definition). We prove two results: (1) if a Kähler manifold satisfies the \(\Delta\)-property then its curvature tensor is parallel; (2) if an Hermitian symmetric space of classical type satisfies the \(\Delta\)-property then it is a complex space form (namely it has constant holomorphic sectional curvature). In view of these results we believe that if a Kähler manifold satisfies the \(\Delta\)-property then it is a complex space form.
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Notes
Namely a Kähler metric admitting a Kähler potential which depends only on the sum \(|z|^2 = |z_1|^2 + \ldots + |z_n|^2\) of the moduli of a local coordinates’ system z (cfr. [5]).
We are going to use the notation \(\partial _i\) to denote \(\frac{\partial }{\partial z_i}\) and a similar notation for higher order derivatives. We are also going to use Einstein’s summation convention for repeated indices.
From now on HSSCT.
From now on HSS.
Namely \(N_i\) is a bounded symmetric domains with a multiple of the Bergman metric denoted by \({\hat{g}}_i\).
References
Alekseevsky, D.V., Perelomov, A.M.: Invariant Kaehler–Einstein metrics on compact homogeneous spaces. Funct. Anal. Appl. 20(3), 171–182 (1986)
Di Scala, A., Loi, A.: Symplectic duality of symmetric spaces. Adv. Math. 217(5), 2336–2352 (2008)
Di Scala, A., Loi, A., Zuddas, F.: Symplectic duality between complex domains. Monatshefte Math. 160(4), 403–428 (2010)
Loi, A., Mossa, R., Zuddas, F.: Bochner coordinates on flag manifolds. Bull. Braz. Math. Soc. (N. S.) 50(2), 497–514 (2019)
Loi, A., Salis, F., Zuddas, F.: On the third coefficient of TYZ expansion for radial scalar flat metrics. J. Geom. Phys. 133, 210–218 (2018)
Lu, Z., Tian, G.: The log term of the Szegő kernel. Duke Math. J. 125(2), 351–387 (2004)
Acknowledgements
The first and the third authors were supported by Prin 2015—Real and Complex Manifolds; Geometry, Topology and Harmonic Analysis—Italy, by GESTA—Funded by Fondazione di Sardegna and Regione Autonoma della Sardegna and by KASBA—Funded by Regione Autonoma della Sardegna. The second author was a research fellow of INdAM. All the three authors were supported by INdAM GNSAGA—Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni.
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Communicated by Vicente Cortés.
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Loi, A., Salis, F. & Zuddas, F. A characterization of complex space forms via Laplace operators. Abh. Math. Semin. Univ. Hambg. 90, 99–109 (2020). https://doi.org/10.1007/s12188-020-00220-0
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DOI: https://doi.org/10.1007/s12188-020-00220-0