Small eigenvalues of the rough and Hodge Laplacians under fixed volume

Colette Anné (1) , Junya Takahashi (2)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 GSIS Mathematics - Research Center for Pure and Applied Mathematics, GSIS, Tohoku University

For each degree p, we construct on any closed manifold a family of Riemannian metrics, with fixed volume such that any positive eigenvalues of the rough and Hodge Laplacians acting on differential p-forms converge to zero. In particular, on the sphere, we can choose these Riemannian metrics as those of non-negative sectional curvature. This is a generalization of the results by Colbois and Maerten in 2010 to the case of higher degree forms. (for a version with the figures see https://www.math.is.tohoku.ac.jp/~junya/article/small-vol-2022.pdf).

Domaines: Théorie spectrale [math.SP]; Géométrie différentielle [math.DG]

Colette Anné, Junya Takahashi. Small eigenvalues of the rough and Hodge Laplacians under fixed volume. 2022. ⟨hal-03268574v2⟩