Eigenvalue and gap estimates for the Laplacian acting on 𝑝-forms

Guerini, Pierre; Savo, Alessandro

doi:10.1090/S0002-9947-03-03336-1

Eigenvalue and gap estimates for the Laplacian acting on $p$-forms

Authors: Pierre Guerini and Alessandro Savo
Journal: Trans. Amer. Math. Soc. 356 (2004), 319-344
MSC (2000): Primary 58J50; Secondary 58J32
DOI: https://doi.org/10.1090/S0002-9947-03-03336-1
Published electronically: August 25, 2003
MathSciNet review: 2020035
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the gap of the first eigenvalue of the Hodge Laplacian acting on $p$-differential forms of a manifold with boundary, for consecutive values of the degree $p$. We first show that the gap may assume any sign. Then we give sufficient conditions on the intrinsic and extrinsic geometry to control it. Finally, we estimate the first Hodge eigenvalue of manifolds whose boundaries have some degree of convexity.

Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
Bruno Colbois and Gilles Courtois, A note on the first nonzero eigenvalue of the Laplacian acting on $p$-forms, Manuscripta Math. 68 (1990), no. 2, 143–160. MR 1063223, DOI https://doi.org/10.1007/BF02568757
Jozef Dodziuk, Eigenvalues of the Laplacian on forms, Proc. Amer. Math. Soc. 85 (1982), no. 3, 437–443. MR 656119, DOI https://doi.org/10.1090/S0002-9939-1982-0656119-2
Leonid Friedlander, Some inequalities between Dirichlet and Neumann eigenvalues, Arch. Rational Mech. Anal. 116 (1991), no. 2, 153–160. MR 1143438, DOI https://doi.org/10.1007/BF00375590
S. Gallot and D. Meyer, Opérateur de courbure et laplacien des formes différentielles d’une variété riemannienne, J. Math. Pures Appl. (9) 54 (1975), no. 3, 259–284 (French). MR 454884
S. Gallot and D. Meyer, D’un résultat hilbertien à un principe de comparaison entre spectres. Applications, Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 4, 561–591 (French). MR 982334
G. Gentile and V. Pagliara, Riemannian metrics with large first eigenvalue on forms of degree $p$, Proc. Amer. Math. Soc. 123 (1995), no. 12, 3855–3858. MR 1277111, DOI https://doi.org/10.1090/S0002-9939-1995-1277111-2

P. Guerini, Spectre du Laplacien de Hodge-de Rham: Estimées sur les Variétés Convexes, Preprint.

P. Guerini, Prescription du Spectre du Laplacien de Hodge-de Rham, Preprint Universität Zürich 2002.

Peter Li and Shing Tung Yau, Estimates of eigenvalues of a compact Riemannian manifold, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 205–239. MR 573435

Jeffrey McGowan, The $p$-spectrum of the Laplacian on compact hyperbolic three manifolds, Math. Ann. 297 (1993), no. 4, 725–745. MR 1245416, DOI https://doi.org/10.1007/BF01459527

H. P. McKean, An upper bound to the spectrum of $\Delta $ on a manifold of negative curvature, J. Differential Geometry 4 (1970), 359–366. MR 266100

L. Payne and H. Weinberger, Lower bounds for vibration frequencies of elastically supportes membranes and plates, J. Soc. Ind. Appl. Math. 5 (1957), 171-182.

Robert C. Reilly, On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space, Comment. Math. Helv. 52 (1977), no. 4, 525–533. MR 482597, DOI https://doi.org/10.1007/BF02567385

A. Savo, A mean-value lemma and applications, Bull. Soc. Math. France 129 (2001), no. 4, 505-542.

Junya Takahashi, On the gap between the first eigenvalues of the Laplacian on functions and 1-forms, J. Math. Soc. Japan 53 (2001), no. 2, 307–320. MR 1815136, DOI https://doi.org/10.2969/jmsj/05320307

J. Takahashi, On the gap between the first eigenvalues of the Laplacian on functions and 1-forms, Ann. Glob. Anal. Geom. to appear.

H.F. Weinberger, An isoperimetric inequality for the n-dimensional free membrane problem, J. Rational Mech. Anal. 5 (1956), 633-636.