Petites valeurs propres des fibrés principaux en tores

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 .