Minakshisundaram–Pleijel zeta function

The Minakshisundaram–Pleijel zeta function is a zeta function encoding the eigenvalues of the Laplacian of a compact Riemannian manifold. It was introduced by Subbaramiah Minakshisundaram and Åke Pleijel (1949). The case of a compact region of the plane was treated earlier by Carleman (1935).

DefinitionEdit

For a compact Riemannian manifold M of dimension N with eigenvalues   of the Laplace–Beltrami operator Δ the zeta function is given for   sufficiently large by

 

(where if an eigenvalue is zero it is omitted in the sum). The manifold may have a boundary, in which case one has to prescribe suitable boundary conditions, such as Dirichlet or Neumann boundary conditions.

More generally one can define

 

for P and Q on the manifold, where the fn are normalized eigenfunctions. This can be analytically continued to a meromorphic function of s for all complex s, and is holomorphic for PQ.

The only possible poles are simple poles at the points s = N/2, N/2−1, N/2−2,..., 1/2,−1/2, −3/2,... for N odd, and at the points s = N/2, N/2−1, N/2−2, ...,2, 1 for N even. If N is odd then Z(P,P,s) vanishes at s = 0, −1, −2,... If N is even. The residues at the poles can be explicitly found in terms of the metric and by Wiener-Ikehara theorem we find as a corollary the relation

 

where the sign ~ indicates that the quotient of both the sides tend to 1 when T tends to +∞ [1].

The function Z(s) can be recovered from Z(P, P, s) by integrating over the whole manifold M:

 

Heat kernelEdit

The analytic continuation of the zeta function can be found by expressing it in terms of the heat kernel

 

as the Mellin transform

 

In particular, we have

 

where

 

is the trace of the heat kernel.

The poles of the zeta function can be found from the asymptotic behavior of the heat kernel as t→0.

ExampleEdit

If the manifold is a circle of dimension N=1, then the eigenvalues of the Laplacian are n2 for integers n. The zeta function

 

where ζ is the Riemann zeta function.

ApplicationsEdit

Apply the method of heat kernel to asymptotic expansion for Riemannian manifold (M,g) we obtain the two following theorems. Both are the resolutions of the inverse problem in which we get the geometric properties or quantities from spectra of the operators.

1) Minakshisundaram-Pleijel Asymptotic Expansion

Let (M,g) be an n-dimensional Riemannian manifold. Then, as t→0+, the trace of the heat kernel has an asymptotic expansion of the form:

 

In dim=2, this means that the integral of scalar curvature tells us the Euler characteristic of M, i.e. Gauss-Bonnet Theorem.

In particular,

 

where S(x) is scalar curvature, the trace of the Ricci curvature, on M.

2) Weyl Asymptotic Formula Let M be a compact Riemannian manifold, with eigenvalues   with each distinct eigenvalue repeated with its multiplicity. Define N(λ) to be the number of eigen values less than or equal to  , and let   denote the volume of the unit disk in  . Then

 

as λ→∞. Additionally, as k→∞,

 

This is also called Weyl's law, refined from Minakshisundaram-Pleijel asymptotic expansion.

ReferencesEdit

  • Berger, Marcel; Gauduchon, Paul; Mazet, Edmond (1971), Le spectre d'une variété riemannienne, Lecture Notes in Mathematics, 194, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0064643, MR 0282313
  • Carleman, Torsten (1935), "Propriétés asymptotiques des fonctions fondamentales des membranes vibrantes.", 8. Skand. Mat.-Kongr. (in French): 34–44, Zbl 0012.07001
  • ^ Minakshisundaram, S.; Pleijel, Å. (1949), "Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds", Canadian Journal of Mathematics, 1: 242–256, doi:10.4153/CJM-1949-021-5, ISSN 0008-414X, MR 0031145