The Minakshisundaram–Pleijel zeta function is a
zeta function
In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function
: \zeta(s) = \sum_^\infty \frac 1 .
Zeta functions include:
* Airy zeta function, related to the zeros of the Airy function
* ...
encoding the eigenvalues of the
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
of a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
. It was introduced by . The case of a compact region of the plane was treated earlier by .
Definition
For a compact Riemannian manifold ''M'' of dimension ''N'' with eigenvalues
of the
Laplace–Beltrami operator
In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named af ...
, 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
Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
or
Neumann boundary condition
In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann.
When imposed on an ordinary or a partial differential equation, the condition specifies the values of the derivative appl ...
s.
More generally one can define
:
for ''P'' and ''Q'' on the manifold, where the
are normalized eigenfunctions. This can be analytically continued to a meromorphic function of ''s'' for all complex ''s'', and is holomorphic for
.
The only possible poles are simple poles at the points
for ''N'' odd, and at the points
for ''N'' even. If ''N'' is odd then
vanishes at
. If ''N'' is even, the residues at the poles can be explicitly found in terms of the metric, and by the
Wiener–Ikehara theorem
The Wiener–Ikehara theorem is a Tauberian theorem introduced by . It follows from Wiener's Tauberian theorem, and can be used to prove the prime number theorem (Chandrasekharan, 1969).
Statement
Let ''A''(''x'') be a non-negative, monotonic no ...
we find as a corollary the relation
:
,
where the symbol
indicates that the quotient of both the sides tend to 1 when T tends to
.
The function
can be recovered from
by integrating over the whole manifold ''M'':
:
.
Heat kernel
The analytic continuation of the zeta function can be found by expressing it in terms of the
heat kernel
In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectru ...
:
as the
Mellin transform In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is
often used i ...
:
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.
Example
If the manifold is a circle of dimension ''N''=1, then the eigenvalues of the Laplacian are ''n''
2 for integers ''n''. The zeta function
:
where ζ is the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
.
Applications
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
In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry ...
tells us the
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
of M, by the
Gauss–Bonnet theorem
In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology.
In the simplest application, the case of a ...
.
In particular,
:
where S(x) is scalar curvature, the trace of the
Ricci curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure ...
, 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 eigenvalues less than or equal to
, and let
denote the volume of the unit disk in
. Then
:
as
. Additionally, as
,
:
This is also called
Weyl's law In mathematics, especially spectral theory, Weyl's law describes the asymptotic behavior of eigenvalues of the Laplace–Beltrami operator. This description was discovered in 1911 (in the d=2,3 case) by Hermann Weyl for eigenvalues for the Laplace ...
, refined from the Minakshisundaram–Pleijel asymptotic expansion.
References
*
*
{{DEFAULTSORT:Minakshisundaram-Pleijel zeta function
Harmonic analysis
Differential geometry
Zeta and L-functions