Abstract
Inspired by the work of Lu and Tian (Duke Math J 125:351--387, 2004), Loi et al. address in (Abh Math Semin Univ Hambg 90: 99-109, 2020) the problem of studying those Kähler manifolds satisfying the \(\Delta \)-property, i.e. such that on a neighborhood of each of its points the k-th power of the Kähler Laplacian is a polynomial function of the complex Euclidean Laplacian, for all positive integer k. In particular they conjectured that if a Kähler manifold satisfies the \(\Delta \)-property then it is a complex space form. This paper is dedicated to the proof of the validity of this conjecture.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Namely a Kähler metric admitting a Kähler potential which depends only on the sum \(|z|^2 = |z_1|^2 + ...+ |z_n|^2\) of the moduli of a local coordinates’ system z.
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.
References
Ballmann, W.: Lectures on Kähler manifolds, ESI Lectures in Mathematics and Physics. Euro Math Soc (EMS), Zürich, (2006) x+172 pp. ISBN: 978-3-03719-025-8; 3-03719-025-6
Di Scala, A., Loi, A.: Symplectic duality of symmetric spaces. Adv. Math. 217(5), 2336–2352 (2008)
Loi, A., Mossa, R.: The diastatic exponential of a symmetric space. Math. Z. 268(3–4), 1057–1068 (2011)
Loi, A., Mossa, R., Zuddas, F.: Symplectic capacities of Hermitian symmetric spaces of compact and noncompact type. J. Sympl. Geom. 13(4), 1049–1073 (2015)
Loi, A., Salis, F., Zuddas, F.: A characterization of complex space forms via Laplace operators. Abh. Math. Semin. Univ. Hambg. 90(1), 99–109 (2020)
Loi, A., Zuddas, F., Roos, G.: The bisymplectomorphism group of a bounded symmetric domain. Transf. Groups 13(2), 283–304 (2008)
Loos, O.: Bounded Symmetric Domains and Jordan pairs. Lecture Notes, University of California, Irvine (1977)
Lu, Z., Tian, G.: The log term of the Szegő kernel. Duke Math. J. 125(2), 351–387 (2004)
Mossa, R.: The volume entropy of local Hermitian symmetric space of noncompact type. Differ. Geom. Appl. 31(5), 594–601 (2013)
Mossa, R., Zedda, M.: Symplectic geometry of Cartan-Hartogs domains, arXiv:2010.05854 [math.DG], (2020)
Roos, G.: Jordan triple systems. In: Faraut, J., Kaneyuki, S., Korányi, A., Lu, Q.K., Roos, G. (eds.) Analysis and Geometry on Complex Homogeneous Domains Progress in Mathematics, vol. 185, pp. 425–534. Birkhäuser, Boston (2000)
Acknowledgements
The author was supported by a Grant from Fapesp (2018/08971-9).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Hr. Cortés.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Mossa, R. On the \(\Delta \)-property for complex space forms. Abh. Math. Semin. Univ. Hambg. 91, 137–143 (2021). https://doi.org/10.1007/s12188-021-00233-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12188-021-00233-3