Small eigenvalues of the rough and Hodge Laplacians under fixed volume

Colette Anné; Junya Takahashi

doi:10.5802/afst.1764

Colette Anné ¹; Junya Takahashi ²

¹ Laboratoire de Mathématiques Jean Leray, Nantes Université, CNRS, Faculté des Sciences, BP 92208, 44322 Nantes, France
² Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tôhoku University, 6-3-09 Aoba, Sendai 980-8579, Japan

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 1, pp. 123-151.

Abstract
Abstract

For each degree $p$ and each natural number $k \geq 1$ , we construct on any closed manifold a family of Riemannian metrics, with fixed volume such that the $k^{th}$ positive eigenvalue of the rough or the Hodge Laplacian acting on differential $p$ -forms converges 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.

Pour chaque degré $p$ et chaque entier naturel $k \geq 1$ , nous construisons, sur toute variété compacte, une famille de métriques riemanniennes à volume fixé telle que la $k^{ième}$ valeur propre strictement positive du Laplacien brut ou du Laplacien de Hodge agissant sur les formes différentielles de degré $p$ converge vers zéro. En particulier, sur la sphère, nous pouvons choisir des métriques à courbure sectionnelle positive ou nulle. Ce résultat généralise aux plus hauts degrés celui de Colbois et Maerten de 2010.

Received: 2021-12-07
Accepted: 2022-02-15
Published online: 2024-07-26

DOI: 10.5802/afst.1764

Classification: 58J50, 35P15, 53C21, 58C40
Keywords: rough Laplacian, Hodge–Laplacian, differential forms, eigenvalues

Author's affiliations:

Colette Anné ¹; Junya Takahashi ²

¹ Laboratoire de Mathématiques Jean Leray, Nantes Université, CNRS, Faculté des Sciences, BP 92208, 44322 Nantes, France
² Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tôhoku University, 6-3-09 Aoba, Sendai 980-8579, Japan

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Colette Anné; Junya Takahashi. Small eigenvalues of the rough and Hodge Laplacians under fixed volume. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 1, pp. 123-151. doi : 10.5802/afst.1764. https://afst.centre-mersenne.org/articles/10.5802/afst.1764/

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