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Curvature and the Eigenvalues of the Laplacian

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Part of the Contemporary Mathematicians book series (CM)

Abstract

A famous formula of H. Weyl [17] states that if D is a bounded region of R d with a piecewise smooth boundary B, and if 0 > γ1 ≥ γ2 ≥ γ3 ≥ etc. is the spectrum of the problem
$$\displaystyle\begin{array}{rcl} \varDelta f =\big (\partial ^{2}/\partial x_{ 1}^{2} + \cdots + \partial ^{2}/\partial x_{ d}^{2}\big)f =\gamma f\quad \mbox{ in }D,& &{}\end{array}$$
(6.1.1a)
$$\displaystyle\begin{array}{rcl} f \in C^{2}(D) \cap C(\overline{D}),& &{}\end{array}$$
(6.1.1b)
$$\displaystyle\begin{array}{rcl} f = 0\quad \mbox{ on }B,& &{}\end{array}$$
(6.1.1c)
then
$$\displaystyle\begin{array}{rcl} -\gamma _{n} \sim C(d)(n/\mbox{ vol }D)^{2/d}\quad (n \uparrow \infty ),& &{}\end{array}$$
(6.1.2)
or, what is the same,
$$\displaystyle\begin{array}{rcl} Z \equiv \mathop{\mathrm{sp}}\nolimits e^{t\varDelta } =\sum _{ n\geq 1}\exp \big(\gamma _{n}t\big) \sim (4\pi t)^{-d/2} \times \mathop{\mathrm{vol}}\nolimits D\quad (t \downarrow 0),& &{}\end{array}$$
(6.1.3)
where \(C(d) = 2\pi [(d/2)!]^{d/2}\).

Keywords

Fundamental Form Elementary Solution Curvature Tensor Riemannian Geometry Closed Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Rockefeller UniversityNew YorkUSA
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA

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