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Petites valeurs propres des fibrés principaux en tores
Author(s) -
Jammes Pierre
Publication year - 2016
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdw010
Subject(s) - mathematics , bounded function , zero (linguistics) , eigenvalues and eigenvectors , laplace operator , dimension (graph theory) , combinatorics , sectional curvature , curvature , mathematical analysis , geometry , scalar curvature , physics , philosophy , linguistics , quantum mechanics
Let M n be a compact manifold of dimension n with free T k ‐action. We consider collapsings of M on N = M / T ksuch that the sectional curvature and diameter of M satisfy | K ( M ) | ⩽ a and diam ( M ) < d , and give examples of collapsings for all k such that the first non‐zero eigenvalue of Laplacian acting on 1‐forms and 2‐forms of M are bounded above by c ( M ) · inj ( M ) 2 k. Moreover, we prove that the first non‐zero eigenvalue of Laplacian acting on 1‐forms of all principal T k ‐bundle M over N is bounded below by c ( n , a , d , N ) · Vol ( M ) 2and c · inj ( M ) 2 kwhen M collapses on N .