A class of 3-dimensional manifolds with bounded first eigenvalue on 1-forms

Gentile, Giovanni

doi:10.1090/S0002-9939-99-04916-3

A class of 3-dimensional manifolds with bounded first eigenvalue on 1-forms
HTML articles powered by AMS MathViewer

by Giovanni Gentile

Proc. Amer. Math. Soc. 127 (1999), 2755-2758

DOI: https://doi.org/10.1090/S0002-9939-99-04916-3

Published electronically: April 23, 1999

PDF | Request permission

Abstract:

Let $(P,g)$ be the framebundle over an oriented, $C^\infty$ Riemannian surface $S$. Denote by $\lambda -{1,1}(P,g)$ the first nonzero eigenvalue of the Laplace operator acting on differential forms of degree 1. We prove that $\lambda _{1,1}(P,g)\le c$ for all $(P,g)$ with canonical metrics $g$ of volume 1.

References

M. Berger, Sur les premières valeurs propres des variétés riemanniennes, Compositio Math. 26 (1973), 129–149 (French). MR 316913
Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684, DOI 10.1007/978-3-540-74311-8
David D. Bleecker, The spectrum of a Riemannian manifold with a unit Killing vector field, Trans. Amer. Math. Soc. 275 (1983), no. 1, 409–416. MR 678360, DOI 10.1090/S0002-9947-1983-0678360-0
Jean-Pierre Bourguignon, Peter Li, and Shing-Tung Yau, Upper bound for the first eigenvalue of algebraic submanifolds, Comment. Math. Helv. 69 (1994), no. 2, 199–207. MR 1282367, DOI 10.1007/BF02564482
B. Colbois and J. Dodziuk, Riemannian metrics with large $\lambda _1$, Proc. Amer. Math. Soc. 122 (1994), no. 3, 905–906. MR 1213857, DOI 10.1090/S0002-9939-1994-1213857-9
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 10.1090/S0002-9939-1995-1277111-2
J. Hersch, Quatre propriétés isopérimetriques des membranes sphériques homogénes, C. R. Acad. Sci. Paris Sér. A 270 (1970), 139–144.
Peter Li and Shing Tung Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69 (1982), no. 2, 269–291. MR 674407, DOI 10.1007/BF01399507
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
L. Polterovich, Symplectic aspects of the first eigenvalue, to appear.
S. Tanno, Geometric expressions of eigen 1-forms of the Laplacian on spheres, Spec. Riem. manifolds, Kaigai Publ. Kyoto (1983), 115–128.
Hajime Urakawa, On the least positive eigenvalue of the Laplacian for compact group manifolds, J. Math. Soc. Japan 31 (1979), no. 1, 209–226. MR 519046, DOI 10.2969/jmsj/03110209
Y. Xu, Diverging Eigenvalues and Collapsing Riemannian Metrics, Institute for Advanced Study, October 1992.
Paul C. Yang and Shing Tung Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), no. 1, 55–63. MR 577325