##
**Spectral asymptotics on degenerating hyperbolic 3-manifolds.**
*(English)*
Zbl 0914.58036

Mem. Am. Math. Soc. 643, 75 p. (1998).

According to Thurston’s hyperbolic surgery theorem, every hyperbolic 3-manifold of finite volume can be approximated by a sequence of compact hyperbolic 3-manifolds. The purpose of the present paper is the study of the behavior under this approximation of certain quantities associated to the Laplace operator acting on functions and to its spectrum. An essential ingredient of all proofs is the “thick and thin” decomposition of hyperbolic manifolds into parts of large versus small injectivity radii. The approximation process is very well-behaved on the thick part; the metrics converge in the \(C^\infty\)-topology. The thin part degenerates but is known very explicitly.

The authors prove convergence of the heat kernels and of some of its derivatives. This uses the explicit formula for the heat kernel on hyperbolic 3-space \(H^3\) and the formula \[ K_M(t,x,y) = \sum_{\gamma\in\Gamma} K_{H^3}(t,\widetilde{x},\gamma\widetilde{y}) \] where \(M=\Gamma\backslash H^3\) is a hyperbolic 3-manifold. When one tries to study the trace of the heat operator, or equivalently integrate \(K_M\) over the diagonal in \(M\times M\), one encounters the problem that the heat operator is not of trace class if the hyperbolic manifold is not compact. Therefore the authors introduce a regularized heat trace in the noncompact case obtained by removing the contribution coming from parabolic elements of \(\Gamma\) (corresponding to the cusps). Then it is possible to prove convergence in the approximation process if one removes the degenerating part from the “compact” heat traces. Similar investigations are performed for Poisson kernels, zeta functions, Selberg zeta functions, Hurwitz type zeta functions, eigenvalue counting functions, and spectral projections.

The methods are an adaption to three dimensions of earlier work by J. Jorgenson and R. Lundelius who investigated the corresponding questions for hyperbolic Riemann surfaces in a series of papers [Quaest. Math. 18, No. 4, 345-363 (1995; Zbl 0853.58099), J. Funct. Anal. 149, No. 1, 25-57 (1997; Zbl 0887.58057), Comment. Math. Helv. 72, No. 4, 636-659 (1997; Zbl 0902.58040)].

The authors prove convergence of the heat kernels and of some of its derivatives. This uses the explicit formula for the heat kernel on hyperbolic 3-space \(H^3\) and the formula \[ K_M(t,x,y) = \sum_{\gamma\in\Gamma} K_{H^3}(t,\widetilde{x},\gamma\widetilde{y}) \] where \(M=\Gamma\backslash H^3\) is a hyperbolic 3-manifold. When one tries to study the trace of the heat operator, or equivalently integrate \(K_M\) over the diagonal in \(M\times M\), one encounters the problem that the heat operator is not of trace class if the hyperbolic manifold is not compact. Therefore the authors introduce a regularized heat trace in the noncompact case obtained by removing the contribution coming from parabolic elements of \(\Gamma\) (corresponding to the cusps). Then it is possible to prove convergence in the approximation process if one removes the degenerating part from the “compact” heat traces. Similar investigations are performed for Poisson kernels, zeta functions, Selberg zeta functions, Hurwitz type zeta functions, eigenvalue counting functions, and spectral projections.

The methods are an adaption to three dimensions of earlier work by J. Jorgenson and R. Lundelius who investigated the corresponding questions for hyperbolic Riemann surfaces in a series of papers [Quaest. Math. 18, No. 4, 345-363 (1995; Zbl 0853.58099), J. Funct. Anal. 149, No. 1, 25-57 (1997; Zbl 0887.58057), Comment. Math. Helv. 72, No. 4, 636-659 (1997; Zbl 0902.58040)].

Reviewer: C.Bär (Freiburg)

### MSC:

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

58J35 | Heat and other parabolic equation methods for PDEs on manifolds |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

57M50 | General geometric structures on low-dimensional manifolds |

35K05 | Heat equation |

58J52 | Determinants and determinant bundles, analytic torsion |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |