Asymptotics of the spectrum of the Laplace operator on Riemannian sol-manifolds in the adiabatic limit. (English. Russian original) Zbl 1213.58019
Sib. Math. J. 51, No. 2, 370-382 (2010); translation from Sib. Mat. Zh. 51, No. 2, 457-472 (2010).
Let \(M_A\) be a 2-torus bundle over \(S^1\) whose monodromy is a hyperbolic linear unimodular map \(A\). The manifold \(M_A\) is a compact quotient of the simply connected \(M_{\text{Sol}}\) model space, Sol being one of the 8 homogeneous 3-dimensional geometries classified by Thurston [see[P. Scott, Bull. Lond. Math. Soc. 15, 401–487 (1983; Zbl 0561.57001)]. The author considers a 1-d foliation \({\mathcal F}\) in \(M_A\) induced by a left invariant vector field on \(M_{\text{Sol}}\) and tangent to the 2-torus fibers: its adiabatic spectral geometry consists of the small \(\varepsilon\) asymptotics of the metric \(g_{{\mathcal F},\varepsilon}=g_{{\mathcal F}}+\varepsilon^{-2} g_{{\mathcal H}}\), where \({\mathcal H}\) is the plane distribution orthogonal to \({\mathcal F}\) and \(g=g_{{\mathcal F}}+g_{{\mathcal H}}\) is the decomposition of the metric \(g\) induced by the orthogonal sum \(TM_A={\mathcal F}\oplus{\mathcal H}\). In particular, we have \(\text{Vol}(M_A,g_{{\mathcal F},\varepsilon})=\text{Vol}(M_A,g)\varepsilon^{-2}\).
Let \(\sigma_{{\mathcal F},\varepsilon}\) be the Laplace-Beltrami operator \(\Delta_{\varepsilon}\) spectrum for the Riemannian manifold \((M_A,g_{{\mathcal F},\varepsilon})\). The main result is the following small \(\varepsilon\) Weyl type asymptotics
\[ \# (\sigma_{{\mathcal F},\varepsilon}\cap[1,t])=C_{\mathcal F} \frac{\text{Vol}(M_A,g)}{6\pi^2\varepsilon^2} {t^{3/2}}(1+o(1)),\quad \varepsilon \to 0^+, \]
where \(t\) is fixed and \(C_{\mathcal F}=1\) if \({\mathcal F} \) is one of the eigendirections of the hyperbolic map \(A\) and \(C_{\mathcal F}=3/2\) otherwise. In the first case, the foliation \({\mathcal F}\) is Riemannian and the asymptotics has been proved also by Y. A. Kordyukov [Math. Ann. 313, No. 4, 763–783 (1999; Zbl 0930.58017)].
Due to the geometry, the eigenvalue equation for the Laplacian \(\Delta_{\varepsilon}\) separates: its spectrum analysis is based on precise spectral asymptotics for the Mathieu hyperbolic operator \(-D_x^2+A\text{cosh}(x)\) on \(\mathbb R\).
These results extend previous work by A. V. Bolsinov, H. R. Dullin and A. P. Veselov [Commun. Math. Phys. 264, No. 3, 583–611 (2006; Zbl 1109.58027)] on the spectrum of the Sol manifold \(M_A\), whose geodesic flow properties have been studied by A. V. Bolsinov and I. A. Taimanov [Invent. Math. 140, No. 3, 639–650 (2000; Zbl 0985.37027)].
Let \(\sigma_{{\mathcal F},\varepsilon}\) be the Laplace-Beltrami operator \(\Delta_{\varepsilon}\) spectrum for the Riemannian manifold \((M_A,g_{{\mathcal F},\varepsilon})\). The main result is the following small \(\varepsilon\) Weyl type asymptotics
\[ \# (\sigma_{{\mathcal F},\varepsilon}\cap[1,t])=C_{\mathcal F} \frac{\text{Vol}(M_A,g)}{6\pi^2\varepsilon^2} {t^{3/2}}(1+o(1)),\quad \varepsilon \to 0^+, \]
where \(t\) is fixed and \(C_{\mathcal F}=1\) if \({\mathcal F} \) is one of the eigendirections of the hyperbolic map \(A\) and \(C_{\mathcal F}=3/2\) otherwise. In the first case, the foliation \({\mathcal F}\) is Riemannian and the asymptotics has been proved also by Y. A. Kordyukov [Math. Ann. 313, No. 4, 763–783 (1999; Zbl 0930.58017)].
Due to the geometry, the eigenvalue equation for the Laplacian \(\Delta_{\varepsilon}\) separates: its spectrum analysis is based on precise spectral asymptotics for the Mathieu hyperbolic operator \(-D_x^2+A\text{cosh}(x)\) on \(\mathbb R\).
These results extend previous work by A. V. Bolsinov, H. R. Dullin and A. P. Veselov [Commun. Math. Phys. 264, No. 3, 583–611 (2006; Zbl 1109.58027)] on the spectrum of the Sol manifold \(M_A\), whose geodesic flow properties have been studied by A. V. Bolsinov and I. A. Taimanov [Invent. Math. 140, No. 3, 639–650 (2000; Zbl 0985.37027)].
Reviewer: Laurent Guillopé (Nantes)
MSC:
| 58J37 | Perturbations of PDEs on manifolds; asymptotics |
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 47A55 | Perturbation theory of linear operators |
| 35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
| 57R30 | Foliations in differential topology; geometric theory |
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