A characterization of complex space forms via Laplace operators

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

  1. 1.

    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 (cfr. [5]).

  2. 2.

    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.

  3. 3.

    From now on HSSCT.

  4. 4.

    From now on HSS.

  5. 5.

    Namely \(N_i\) is a bounded symmetric domains with a multiple of the Bergman metric denoted by \({\hat{g}}_i\).

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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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Correspondence to Andrea Loi.

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Loi, A., Salis, F. & Zuddas, F. A characterization of complex space forms via Laplace operators. Abh. Math. Semin. Univ. Hambg. (2020). https://doi.org/10.1007/s12188-020-00220-0

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Keywords

  • Kähler manifolds
  • Hermitian symmetric spaces
  • Kähler Laplacian

Mathematics Subject Classification

  • Primary 53C55
  • Secondary 58C25
  • 58F06