Chapter 1 DIFFERENTIAL GEOMETRY OF REAL MANIFOLDS

290 CHAPTER 1. DIFFERENTIAL GEOMETRY ...
formula for N(λ) to a compact Riemannian manifold M (with or without boundary) but the
above method based on the variational characterization does not seem to generalize. For the
remainder R(λ) = N(λ) − vmvol(M)
(2π) m λ m
2 we have
R(λ) = O(λ m−1
2 ).
That this estimate for the remainder is sharp can be established by elementary arguments
using the explicit knowledge of eigenvalues on the sphere, however the proof . It is remarkable
that the remainder is related to the existence of periodic geodesics on M. In fact on manifolds
where the geodesic flow is not periodic the estimate can be improved. For a discussion of the
remainder the reader is referred to [Ho] and [DG]. For manifolds with boundary the standard
conjecture for the asymptotic distribution of eigenvalues was
N(λ) = vmvol(M) m
λ 2 −
(2π) m cmvol(∂M)
(2π)
m−1
λ 2 + o(λ m−1 m−1
where cm is a constant depending only on m. That this formula is not valid was established
by R. Melrose et al. For an account of N(λ) for Riemannian manifolds with boundary see
[Iv], [Pet] and references thereof.
Note that the analysis in the finite dimensional case is basic linear algebra. To make a
story in the finite dimensional case, we replace the compact manifold M with a finite graph.
Let V be the set of vertices and E the set of edges of a finite graph Γ (with no loops, i.e., an
edge joining a vertex to itself; and no multiple edges). If two vertices u, v are connected an
edge, we write u ↔ v. For v ∈ V let δv denote the number of vertices u such that v ↔ u.
Let L denote the set of real or complex valued functions on V which is a finite dimensional
vector space. One may define the Laplacian on L as
∆ϕ(u) = −ϕ(u) + 1
√ δu
v,v↔u
ϕ(v)
√ .
δv
With this definition (or some generalizations of it) on may transport a portion of the theory
of the Laplacian in differential geometry or analysis to the context of graphs and Markov
chains. For a discussion of this aspect of the subject see [Chu].
Example 1.3.8 While the eigenvalue λ1 > 0, exercise 1.3.24 shows that it can be arbitrarily
small by taking γ large. We now give a class of examples of compact surfaces for which λ1 > 0
is arbitrarily small and sheds some light on how to obtain a lower bound for λ1 which depends
on geometric data. Let Mi, i = 1, 2, be a compact surfaces with Riemannian metrics ds 2 i .
Assume there are small discs Dj ⊂ Mj where ds 2 j is flat. Join the surfaces by a cylinder of P
2 ,