Curvature and the Eigenvalues of the Laplacian

Part of the Contemporary Mathematicians book series (CM)


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
$$\displaystyle\begin{array}{rcl} \varDelta f =\big (\partial ^{2}/\partial x_{ 1}^{2} + \cdots + \partial ^{2}/\partial x_{ d}^{2}\big)f =\gamma f\quad \mbox{ in }D,& &{}\end{array}$$
$$\displaystyle\begin{array}{rcl} f \in C^{2}(D) \cap C(\overline{D}),& &{}\end{array}$$
$$\displaystyle\begin{array}{rcl} f = 0\quad \mbox{ on }B,& &{}\end{array}$$
$$\displaystyle\begin{array}{rcl} -\gamma _{n} \sim C(d)(n/\mbox{ vol }D)^{2/d}\quad (n \uparrow \infty ),& &{}\end{array}$$
or, what is the same,
$$\displaystyle\begin{array}{rcl} Z \equiv \mathop{\mathrm{sp}}\nolimits e^{t\varDelta } =\sum _{ n\geq 1}\exp \big(\gamma _{n}t\big) \sim (4\pi t)^{-d/2} \times \mathop{\mathrm{vol}}\nolimits D\quad (t \downarrow 0),& &{}\end{array}$$
where \(C(d) = 2\pi [(d/2)!]^{d/2}\).


Fundamental Form Elementary Solution Curvature Tensor Riemannian Geometry Closed Manifold 
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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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