Pour chaque degré et chaque entier naturel , nous construisons, sur toute variété compacte, une famille de métriques riemanniennes à volume fixé telle que la valeur propre strictement positive du Laplacien brut ou du Laplacien de Hodge agissant sur les formes différentielles de degré 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.
For each degree and each natural number , we construct on any closed manifold a family of Riemannian metrics, with fixed volume such that the positive eigenvalue of the rough or the Hodge Laplacian acting on differential -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.
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Mots clés : rough Laplacian, Hodge–Laplacian, differential forms, eigenvalues
Colette Anné 1 ; Junya Takahashi 2
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@article{AFST_2024_6_33_1_123_0, author = {Colette Ann\'e and Junya Takahashi}, title = {Small eigenvalues of the rough and {Hodge} {Laplacians} under fixed volume}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {123--151}, publisher = {Universit\'e Paul Sabatier, Toulouse}, volume = {Ser. 6, 33}, number = {1}, year = {2024}, doi = {10.5802/afst.1764}, language = {en}, url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1764/} }
TY - JOUR AU - Colette Anné AU - Junya Takahashi TI - Small eigenvalues of the rough and Hodge Laplacians under fixed volume JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2024 SP - 123 EP - 151 VL - 33 IS - 1 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1764/ DO - 10.5802/afst.1764 LA - en ID - AFST_2024_6_33_1_123_0 ER -
%0 Journal Article %A Colette Anné %A Junya Takahashi %T Small eigenvalues of the rough and Hodge Laplacians under fixed volume %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2024 %P 123-151 %V 33 %N 1 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1764/ %R 10.5802/afst.1764 %G en %F AFST_2024_6_33_1_123_0
Colette Anné; Junya Takahashi. Small eigenvalues of the rough and Hodge Laplacians under fixed volume. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 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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