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Estimations spectrales asymptotiques en géométrie hermitienne

Abstract : The goal of this thesis is to study several differential geometry problems within complex and almost complex frameworks. First, we give some formulas of the Bochner-Kodaira-Nakano-type for Hermitian vector bundles over Hermitian, almost Kähler and respectively almost complex manifolds. Then, by means of one of the previous formulas, we obtain asymptotical estimates of a portion of the spectrum of certain differential operators in the complex case; more precisely, starting from a closed real (not necessarily entire) $(1,1)$-form $\alpha$ on a compact complex manifold of dimension $n$, we construct a sequence of Hermitian line bundles, indexed over $k$, whose curvature forms approximate $k\alpha$. The asymptotic estimates deal with the lower part of the spectrum of the antiholomorphic Laplacians associated with these vector bundles, while the most significant of them involves the integral of $\alpha^n$ over the points of index 0 or 1 in the manifold. This is only significant if the last-mentioned integral is positive.
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https://tel.archives-ouvertes.fr/tel-00002098
Contributor : Arlette Guttin-Lombard <>
Submitted on : Monday, December 9, 2002 - 4:28:34 PM
Last modification on : Wednesday, November 4, 2020 - 1:58:04 PM
Long-term archiving on: : Tuesday, September 11, 2012 - 6:50:39 PM

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• HAL Id : tel-00002098, version 1

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Laurent Laeng. Estimations spectrales asymptotiques en géométrie hermitienne. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2002. Français. ⟨tel-00002098⟩

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