From vanishing theorems to estimating theorems: the Bochner technique revisited

DOI10.1090/S0273-0979-1988-15679-0zbMath0662.53037MaRDI QIDQ3811308

Pierre H. Bérard

Publication date: 1988

Published in: Bulletin of the American Mathematical Society (Search for Journal in Brave)

zbMATH Keywords

maximum principleheat equationLaplacianvanishing theoremsWeitzenböck formulasBochner techniqueestimating theoremsKato inequalities

Mathematics Subject Classification ID

Spectral problems; spectral geometry; scattering theory on manifolds (58J50) Global Riemannian geometry, including pinching (53C20) Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) (58J60) Research exposition (monographs, survey articles) pertaining to global analysis (58-02) Research exposition (monographs, survey articles) pertaining to differential geometry (53-02) Heat and other parabolic equation methods for PDEs on manifolds (58J35)

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Cites Work

• Unnamed Item
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• Unnamed Item
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• Unnamed Item
• Unnamed Item
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• Unnamed Item
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• Unnamed Item
• Unnamed Item
• Unnamed Item
• Pinching and Betti numbers
• Positive scalar curvature and the Dirac operator on complete Riemannian manifolds
• Symétrisation d'inéquations elliptiques et applications géométriques. (Symmetrization of elliptic inequalities and geometric applications)
• Sur une inégalité isopérimétrique qui généralise celle de Paul Lévy-Gromov. (On an isoperimetric inequality which generalizes that of Paul Lévy-Gromov)
• Vanishing theorems on complex manifolds
• Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes
• Kato's inequality and the spectral distribution of Laplacians on compact Riemannian manifolds
• Curvature, diameter and Betti numbers
• Vanishing theorems for square-integrable harmonic forms
• Opérateur de courbure et laplacien des formes différentielles d'une variété riemannienne
• Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire
• Best constant in Sobolev inequality
• Harmonic spinors
• On stability of minimal spheres and a two-dimensional version of Synge's theorem
• Le spectre d'une variété riemannienne. (The spectrum of a Riemannian manifold)
• On Automorphisms of A Kählerian Structure
• Inégalités isopérimétriques et applications
• Sur le volume minimal de ${R}\sp 2$
• Stability of Harmonic Maps and Eigenvalues of the Laplacian
• Some isoperimetric inequalities and eigenvalue estimates
• On the Sobolev constant and the $p$-spectrum of a compact riemannian manifold
• Seminar on Differential Geometry. (AM-102)
• Elliptic Partial Differential Equations of Second Order
• Curvature and Betti Numbers. (AM-32)
• Vector fields and Ricci curvature