Nondegeneracy of the eigenvalues of the Hodge Laplacian for generic metrics on 3-manifolds. (English) Zbl 1286.58021
Let \(M\) be a compact three-dimensional Riemannian manifold and \(r\geq 2.\) It is the main result of the paper that there is a residual subset \(\Gamma\) of the space of \(C^r\)-metrics such that, for all \(g\in \Gamma,\) the following holds: The non-zero eigenvalues of the Hodge Laplacian \(\Delta_g\) on \(p\)-forms have multiplicity \(1\) for \(0 \leq p\leq 3\) and the zero set of eigenforms of degree \(p=1,2\) are isolated. The authors use a detailed analysis of the Beltrami operator \(\star_g \circ d.\)
Reviewer: Hans-Bert Rademacher (Leipzig)
MSC:
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 35B38 | Critical points of functionals in context of PDEs (e.g., energy functionals) |
References:
| [1] | Jeffrey H. Albert, Generic properties of eigenfunctions of elliptic partial differential operators, Trans. Amer. Math. Soc. 238 (1978), 341 – 354. · Zbl 0379.35023 |
| [2] | Colette Anné and Bruno Colbois, Spectre du laplacien agissant sur les \?-formes différentielles et écrasement d’anses, Math. Ann. 303 (1995), no. 3, 545 – 573 (French). · Zbl 0909.58054 · doi:10.1007/BF01461004 |
| [3] | N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl. (9) 36 (1957), 235 – 249. · Zbl 0084.30402 |
| [4] | Shigetoshi Bando and Hajime Urakawa, Generic properties of the eigenvalue of the Laplacian for compact Riemannian manifolds, Tôhoku Math. J. (2) 35 (1983), no. 2, 155 – 172. · Zbl 0534.58038 · doi:10.2748/tmj/1178229047 |
| [5] | Christian Bär, Zero sets of solutions to semilinear elliptic systems of first order, Invent. Math. 138 (1999), no. 1, 183 – 202. · Zbl 0942.35064 · doi:10.1007/s002220050346 |
| [6] | David D. Bleecker and Leslie C. Wilson, Splitting the spectrum of a Riemannian manifold, SIAM J. Math. Anal. 11 (1980), no. 5, 813 – 818. · Zbl 0449.58021 · doi:10.1137/0511072 |
| [7] | Sagun Chanillo and François Treves, On the lowest eigenvalue of the Hodge Laplacian, J. Differential Geom. 45 (1997), no. 2, 273 – 287. · Zbl 0874.58087 |
| [8] | Mircea Craioveanu, Mircea Puta, and Themistocles M. Rassias, Old and new aspects in spectral geometry, Mathematics and its Applications, vol. 534, Kluwer Academic Publishers, Dordrecht, 2001. · Zbl 0987.58013 |
| [9] | Mattias Dahl, Dirac eigenvalues for generic metrics on three-manifolds, Ann. Global Anal. Geom. 24 (2003), no. 1, 95 – 100. · Zbl 1035.53065 · doi:10.1023/A:1024231524848 |
| [10] | John Etnyre and Robert Ghrist, Contact topology and hydrodynamics. III. Knotted orbits, Trans. Amer. Math. Soc. 352 (2000), no. 12, 5781 – 5794. · Zbl 0960.76020 |
| [11] | John Etnyre and Robert Ghrist, Generic hydrodynamic instability of curl eigenfields, SIAM J. Appl. Dyn. Syst. 4 (2005), no. 2, 377 – 390. · Zbl 1088.76012 · doi:10.1137/04060651X |
| [12] | Robert Hardt and Leon Simon, Nodal sets for solutions of elliptic equations, J. Differential Geom. 30 (1989), no. 2, 505 – 522. · Zbl 0692.35005 |
| [13] | Dmitry Jakobson and Nikolai Nadirashvili, Eigenfunctions with few critical points, J. Differential Geom. 53 (1999), no. 1, 177 – 182. · Zbl 1038.58036 |
| [14] | Pierre Jammes, Construction de valeurs propres doubles du laplacien de Hodge-de Rham, J. Geom. Anal. 19 (2009), no. 3, 643 – 654 (French, with English summary). · Zbl 1173.58011 · doi:10.1007/s12220-009-9079-6 |
| [15] | Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. · Zbl 0836.47009 |
| [16] | Jerry L. Kazdan, Unique continuation in geometry, Comm. Pure Appl. Math. 41 (1988), no. 5, 667 – 681. · Zbl 0632.35015 · doi:10.1002/cpa.3160410508 |
| [17] | John Lott, Collapsing and the differential form Laplacian: the case of a smooth limit space, Duke Math. J. 114 (2002), no. 2, 267 – 306. · Zbl 1072.58023 · doi:10.1215/S0012-7094-02-11424-0 |
| [18] | Tatiana Mantuano, Discretization of Riemannian manifolds applied to the Hodge Laplacian, Amer. J. Math. 130 (2008), no. 6, 1477 – 1508. · Zbl 1158.58015 · doi:10.1353/ajm.0.0025 |
| [19] | Richard S. Millman, Remarks on spectrum of the Laplace-Beltrami operator in the middle dimensions, Tensor (N.S.) 34 (1980), no. 1, 94 – 96. · Zbl 0436.53044 |
| [20] | Frank Quinn, Transversal approximation on Banach manifolds, Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 213 – 222. |
| [21] | K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. Math. 98 (1976), no. 4, 1059 – 1078. · Zbl 0355.58017 · doi:10.2307/2374041 |
| [22] | Shing Tung Yau , Seminar on Differential Geometry, Annals of Mathematics Studies, vol. 102, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. Papers presented at seminars held during the academic year 1979 – 1980. · Zbl 0471.00020 |
| [23] | Shing-Tung Yau, Open problems in geometry, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 1 – 28. · Zbl 0801.53001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
