Spectral gap estimates on compact manifolds

Oden, Kevin; Sung, Chiung-Jue; Wang, Jiaping

doi:10.1090/S0002-9947-99-02039-5

Spectral gap estimates on compact manifolds
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by Kevin Oden, Chiung-Jue Sung and Jiaping Wang PDF

Trans. Amer. Math. Soc. 351 (1999), 3533-3548 Request permission

Abstract:

For a compact Riemannian manifold with boundary, its mass gap is the difference between the first and second smallest Dirichlet eigenvalues. In this paper, taking a variational approach, we obtain an explicit lower bound estimate of the mass gap for any compact manifold in terms of geometric quantities.

References

Herm Jan Brascamp and Elliott H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis 22 (1976), no. 4, 366–389. MR 0450480, DOI 10.1016/0022-1236(76)90004-5
Michael E. Gage, Upper bounds for the first eigenvalue of the Laplace-Beltrami operator, Indiana Univ. Math. J. 29 (1980), no. 6, 897–912. MR 589652, DOI 10.1512/iumj.1980.29.29061
Sylvestre Gallot, Dominique Hulin, and Jacques Lafontaine, Riemannian geometry, 2nd ed., Universitext, Springer-Verlag, Berlin, 1990. MR 1083149, DOI 10.1007/978-3-642-97242-3
David Jerison, The Poincaré inequality for vector fields satisfying Hörmander’s condition, Duke Math. J. 53 (1986), no. 2, 503–523. MR 850547, DOI 10.1215/S0012-7094-86-05329-9
Roger Chen, Neumann eigenvalue estimate on a compact Riemannian manifold, Proc. Amer. Math. Soc. 108 (1990), no. 4, 961–970. MR 993745, DOI 10.1090/S0002-9939-1990-0993745-X
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
L. Saloff-Coste and D. W. Stroock, Opérateurs uniformément sous-elliptiques sur les groupes de Lie, J. Funct. Anal. 98 (1991), no. 1, 97–121 (French, with English summary). MR 1111195, DOI 10.1016/0022-1236(91)90092-J
I. M. Singer, Bun Wong, Shing-Tung Yau, and Stephen S.-T. Yau, An estimate of the gap of the first two eigenvalues in the Schrödinger operator, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 2, 319–333. MR 829055
Ernst Heintze and Hermann Karcher, A general comparison theorem with applications to volume estimates for submanifolds, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 451–470. MR 533065, DOI 10.24033/asens.1354
F. W. Warner, Extensions of the Rauch comparison theorem to submanifolds, Trans. Amer. Math. Soc. 122 (1966), 341–356. MR 200873, DOI 10.1090/S0002-9947-1966-0200873-6
S.Y. Cheng, K. Oden, Isoperimetric Inequalities and the Gap Between the First and Second Eigenvalues of an Euclidean domain, preprint.
K. Oden, UCLA dissertation, 1994
R. Schoen, S.T. Yau, Differential Geometry, International Press (1995).
Jiaping Wang, Global heat kernel estimates, Pacific J. Math. 178 (1997), no. 2, 377–398. MR 1447421, DOI 10.2140/pjm.1997.178.377