Adiabatic limits and the spectrum of the Laplacian on foliated manifolds.

*(English)*Zbl 1159.58018
Burghelea, Dan (ed.) et al., \(C^*\)-algebras and elliptic theory II. Selected papers of the international conference, Bȩdlewo, Poland, January 2006. Basel: Birkhäuser (ISBN 978-3-7643-8603-0/hbk). Trends in Mathematics, 123-144 (2008).

Let \(\mathcal{F}\) be a smooth foliation on a manifold \(M\) with a Riemannian metric \(g\). The tangent bundle decomposes as direct sum of the subbundles of vectors tangent and perpendicular to the leaves, respectively: \(TM=F\oplus H\). This yields a corresponding decomposition of the metric tensor, \(g=g_F+g_H\), and we can consider the family of metrics \(g_h=g_F+h^{-2}g_H\) for \(h>0\). The study of the asymptotic behaviour of \((M,g_h)\) as \(h\to0\), called adiabatic limit, was begun by E. Witten [Commun. Math. Phys. 100, 197–229 (1985; Zbl 0581.58038)], and continued by many authors.

In the present paper, the authors consider the adiabatic limit of the spectrum of the Laplacian as follows. Let \(\Delta_h\) denote the laplacian defined by \(g_h\), and let \[ 0\leq\lambda_0^r(h)\leq\lambda_1^r(h)\leq\lambda_2^r(h)\leq\cdots \] be its spectrum on \(r\)-forms (with multiplicities); each function \(\lambda_i^r(h)\) is called a spectral branch. The main results about the asymptotics of the spectral branches as \(h\to0\) were achieved when \(\mathcal{F}\) is Riemannian and \(g\) bundle-like (\(g_H\) is “leafwise constant”). For instance, given a Schwartz function \(f\) on \(\mathbb{R}\), a version of the semiclassical Weil formula was proved by the first author in [Math. Ann. 313, 763–783 (1999; Zbl 0930.58017], describing the asymptotics of \(f(\Delta_h)\) as \(h\to0\) when \(\mathcal{F}\) is Riemannian and \(g\) bundle-like. Under these conditions, it was also shown by the first author and the reviewer [Geom. Funct. Anal. 10, 977–1027 (2000; Zbl 0965.57024] that the spectral branches converging to zero as \(h\to0\), called small spectrum, are related to the spectral sequence \(E_k\) of \(\mathcal{F}\), which is the obvious generalization of the Leray spectral sequence of fiber bundles defined by filtering the de Rham complex.

The paper under review is a presentation of the above results and other recent results whose detailed proofs are in two preprints of the second author and his publication [Sb. Math. 199, No. 2, 307–318 (2008); translation from Mat. Sb. 199, No. 2, 149–160 (2008; Zbl 1157.58009)]. These new results provide very concrete computations of the spectral branches in two examples. In the first one, \(M\) is the \(2\)-torus \(\mathbb{R}^2/\mathbb{Z}^2\), \(\mathcal{F}\) is the Kronecker flow defined by lines of a fixed slope \(\alpha\) on \(\mathbb{R}^2\), and \(g\) is the flat metric. In particular, when \(\alpha\not\in\mathbb{Q}\), it is shown that the spectral distribution function \(N_h(\lambda)\) of \(\Delta_h\) on functions has the following asymptotic expression when \(h\to0\): \[ N_h(\lambda)=\frac{1}{4\pi}h^{-1}\lambda+o(h^{-1})\;. \] The second example is very interesting because the foliation is not Riemannian: \((M,g)\) is a three-dimensional Riemannian Heisenberg manifold \(\Gamma\backslash H\), and \(\mathcal{F}\) is given by a left invariant vector field on \(H\) which is not in the center of its Lie algebra. In this case, the computation of the spectrum of the Laplacian on functions was made by C. S. Gordon and E. N. Wilson [Mich. Math. J. 33, 253–271 (1986; Zbl 0599.53038)]. The authors compute the spectrum also on forms, obtaining the following asymptotic formula as \(h\to 0\) for the trace of the heat operator: \[ \text{tr}e^{-t\Delta_h}=\frac{h^{-2}}{4\pi t}\int_{-\infty}^{+\infty}\frac{\eta}{\sinh(t\eta)}\,e^{-t\eta^2}\,d\eta+o(h^{-2})\;. \] Here, the factor \(\frac{\eta}{\sinh(t\eta)}\) is given by the leafwise variation of \(g_H\); thus this factor would not show up in the case of a Riemannian foliation. Nevertheless it is shown that the small spectrum of this example is related to the spectral sequence \(E_k\) like in the case of Riemannian foliations.

For the entire collection see [Zbl 1134.58002].

In the present paper, the authors consider the adiabatic limit of the spectrum of the Laplacian as follows. Let \(\Delta_h\) denote the laplacian defined by \(g_h\), and let \[ 0\leq\lambda_0^r(h)\leq\lambda_1^r(h)\leq\lambda_2^r(h)\leq\cdots \] be its spectrum on \(r\)-forms (with multiplicities); each function \(\lambda_i^r(h)\) is called a spectral branch. The main results about the asymptotics of the spectral branches as \(h\to0\) were achieved when \(\mathcal{F}\) is Riemannian and \(g\) bundle-like (\(g_H\) is “leafwise constant”). For instance, given a Schwartz function \(f\) on \(\mathbb{R}\), a version of the semiclassical Weil formula was proved by the first author in [Math. Ann. 313, 763–783 (1999; Zbl 0930.58017], describing the asymptotics of \(f(\Delta_h)\) as \(h\to0\) when \(\mathcal{F}\) is Riemannian and \(g\) bundle-like. Under these conditions, it was also shown by the first author and the reviewer [Geom. Funct. Anal. 10, 977–1027 (2000; Zbl 0965.57024] that the spectral branches converging to zero as \(h\to0\), called small spectrum, are related to the spectral sequence \(E_k\) of \(\mathcal{F}\), which is the obvious generalization of the Leray spectral sequence of fiber bundles defined by filtering the de Rham complex.

The paper under review is a presentation of the above results and other recent results whose detailed proofs are in two preprints of the second author and his publication [Sb. Math. 199, No. 2, 307–318 (2008); translation from Mat. Sb. 199, No. 2, 149–160 (2008; Zbl 1157.58009)]. These new results provide very concrete computations of the spectral branches in two examples. In the first one, \(M\) is the \(2\)-torus \(\mathbb{R}^2/\mathbb{Z}^2\), \(\mathcal{F}\) is the Kronecker flow defined by lines of a fixed slope \(\alpha\) on \(\mathbb{R}^2\), and \(g\) is the flat metric. In particular, when \(\alpha\not\in\mathbb{Q}\), it is shown that the spectral distribution function \(N_h(\lambda)\) of \(\Delta_h\) on functions has the following asymptotic expression when \(h\to0\): \[ N_h(\lambda)=\frac{1}{4\pi}h^{-1}\lambda+o(h^{-1})\;. \] The second example is very interesting because the foliation is not Riemannian: \((M,g)\) is a three-dimensional Riemannian Heisenberg manifold \(\Gamma\backslash H\), and \(\mathcal{F}\) is given by a left invariant vector field on \(H\) which is not in the center of its Lie algebra. In this case, the computation of the spectrum of the Laplacian on functions was made by C. S. Gordon and E. N. Wilson [Mich. Math. J. 33, 253–271 (1986; Zbl 0599.53038)]. The authors compute the spectrum also on forms, obtaining the following asymptotic formula as \(h\to 0\) for the trace of the heat operator: \[ \text{tr}e^{-t\Delta_h}=\frac{h^{-2}}{4\pi t}\int_{-\infty}^{+\infty}\frac{\eta}{\sinh(t\eta)}\,e^{-t\eta^2}\,d\eta+o(h^{-2})\;. \] Here, the factor \(\frac{\eta}{\sinh(t\eta)}\) is given by the leafwise variation of \(g_H\); thus this factor would not show up in the case of a Riemannian foliation. Nevertheless it is shown that the small spectrum of this example is related to the spectral sequence \(E_k\) like in the case of Riemannian foliations.

For the entire collection see [Zbl 1134.58002].

##### MSC:

58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |

57R30 | Foliations in differential topology; geometric theory |