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Transactions of the American Mathematical Society

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On the Laplace-Beltrami operator on compact complex spaces


Author: Francesco Bei
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 32W05, 32W50, 35P15; Secondary 58J35
DOI: https://doi.org/10.1090/tran/7848
Published electronically: May 30, 2019
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Abstract: Let $ (X,h)$ be a compact and irreducible Hermitian complex space of complex dimension $ v>1$. In this paper we show that the Friedrichs extension of both the Laplace-Beltrami operator and the Hodge-Kodaira Laplacian acting on functions has discrete spectrum. Moreover, we provide some estimates for the growth of the corresponding eigenvalues, and we use these estimates to deduce that the associated heat operators are trace class. Finally we give various applications to the Hodge-Dolbeault operator and to the Hodge-Kodaira Laplacian in the setting of Hermitian complex spaces of complex dimension $ 2$.


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Additional Information

Francesco Bei
Affiliation: Dipartimento di Matematica, Università degli Studi di Padova, Padova, Italy
Email: bei@math.unipd.it; francescobei27@gmail.com

DOI: https://doi.org/10.1090/tran/7848
Keywords: Hermitian complex space, Laplace--Beltrami operator, Sobolev inequality, $\overline{\partial}$-operator, Hodge--Kodaira Laplacian, complex surface
Received by editor(s): April 22, 2018
Received by editor(s) in revised form: March 13, 2019
Published electronically: May 30, 2019
Additional Notes: This work was performed within the framework of the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR)
Article copyright: © Copyright 2019 American Mathematical Society