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).

Definition

For a compact Riemannian manifold M of dimension N with eigenvalues $\lambda_1, \lambda_2, \ldots$ of the Laplace–Beltrami operator Δ the zeta function is given for $\operatorname{Re}(s)$ sufficiently large by

$Z(s) = \operatorname{Tr}(\Delta^{-s}) = \sum_{n=1}^{\infty} \vert \lambda_{n} \vert^{-s}.$

(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

$Z(P, Q, s) = \sum_{n=1}^{\infty} \frac{f_n(P)f_n(Q)}{ \lambda_{n}^s}$

for P and Q on the manifold, where the f_n are normalized eigenfunctions. This can be analytically continued to a meromorphic function of s for all complex s, and is holomorphic for P≠Q. 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,Q,s) vanishes for s = 0, −1, −2,... The function Z(s) can be recovered from this by integrating Z(P, P, s) over the whole manifold M:

$\displaystyle Z(s) = \int_M Z(P,P,s)dP$

Heat kernel

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

$K(P,Q,t) = \sum_{n=1}^{\infty} f_n(P)f_n(Q) e^{- \lambda_{n}t}$

as the Mellin transform

$Z(P,Q,s) = \frac{1}{\Gamma(s)} \int_0^\infty K(P,Q,t) t^{s-1} dt$

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

Example

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

$Z(s) = \sum_{n\ne 0}\frac{1}{(n^2)^s} = 2\zeta(2s)$

where ζ is the Riemann zeta function.

References

• 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, MR0282313 
• Carleman, Torsten (1935), "Propriétés asymptotiques des fonctions fondamentales des membranes vibrantes." (in French), 8. Skand. Mat.-Kongr.: 34–44, Zbl 0012.07001, http://books.google.com/books?id=ZdHGSgAACAAJ 
• 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, MR0031145, http://math.ca/10.4153/CJM-1949-021-5