Laplacian Eigenvalues and Distances Between Subsets of a Manifold

Joel Friedman^{∗} Department of Mathematics University of British Colubmia

Vancouver, BC V6T 1Z2 CANADA

        jf@math.ubc.ca

Jean-Pierre Tillich LRI bâtiment 490 Université Paris-Sud Orsay 91405 FRANCE

                                 tillich@lri.fr

November 22, 2000

Abstract

In this paper we give a new method to convert results dealing with graph theoretic (or Markov chain) Laplacians into results concerning Laplacians in analysis, such as on Riemannian manifolds. We illustrate this method by using the results of [CGY97] to prove

λ_{1} ≤ \frac{1}{dist^{2}(X, Y)} (cosh^{−1} \sqrt{\frac{μX^{c} μY^{c}}{μX μY}} )^{2} .

for λ_{1} the first positive Neumann eigenvalue on a connected compact Riemannian manifold, and X, Y any two disjoint sets (and where X^{c} is the complement of X). This inequality has a version for the k-th positive eigenvalue (involving k+1 disjoint sets), and holds more generally for all “analytic” Laplacians described in [CGY97]. We show that this inequality is optimal “to first order,” in that it is impossible to obtain an inequality of this form with the right-hand-side divided by 1 + ϵ for any fixed constant ϵ > 0.

1 Introduction

This papers gives a new method for converting results on discrete math Laplacians, e.g. Laplacians on graphs or Markov chains, into results on analysis Laplacians, e.g.

∗Research supported in part by an NSERC grant.

Laplacians on Riemannian manifolds. Converting in the reverse direction is found in [FT], whose methods partially lead to the method in this paper. We shall illustrate our method on one class of examples, which we now describe.

To fix ideas, let M be a compact connected Riemannian manifold, possibly with boundary, and λ_{1} the smallest positive Neumann eigenvalue of the Laplacian. Then if X, Y are disjoint Borel sets in M, Chung, Grigor’yan, and Yau, in [CGY96], showed that

λ_{1} ≤ \frac{1}{dist^{2}(X, Y)} (log \frac{C(μM)^{2}}{μX μY} )^{2}  (1.1)

with C = 4, where μ is the Riemannian volume measure (and where the logarithm is to base e). Later, in [CGY97], they obtained inequality 1.1 with C = e, and Bobkov and Ledoux, in [BL97], obtained C = 1. We mention that in [CGY97] the above inequality holds for more general Riemannian manifolds and more general Laplacians, and in [BL97] holds in the even greater generality of probability metric spaces^{1}. Also, we mention that such inequalities have appeared in the context of graphs and certain metric probability spaces, as in [AM85, MS86, Moh91, LPS88, Sar90, CFM94, CGY97] and the references there (specifically for graphs see [AM85] theorem 2.6 with a = 1/2,

[Moh91] lemma 2.4, and [CGY97]).

In this paper we will prove the theorem below.

Theorem 1.1 Let L be an analytic Laplacian on a metric measure space (M, μ), and let λ_{1} be the lower bound of the spectrum of L acting on the functions orthogonal to the constants. Then for X, Y disjoint Borel subsets of M we have

λ_{1} ≤ \frac{1}{dist^{2}(X, Y)} (cosh^{−1} \sqrt{\frac{μX^{c} μY^{c}}{μX μY}} )^{2} ,

where X^{c} is the complement of X.

This theorem will, in particular, apply to the second Neumann eigenvalue of a compact Riemannian manifold as above. In particular, using cosh^{−1}(y) ≤ log(2y) and that μX^{c}, μY^{c} ≤ μM, we conclude:

λ_{1} ≤ \frac{1/4}{dist^{2}(X, Y)} (log \frac{4(μM)^{2}}{μX μY} )^{2} .  (1.2)

Furthermore we give a simple class of examples to show that any inequality of the form

λ_{1} ≤ \frac{C_{1}}{dist^{2}(X, Y)} (log \frac{C_{2}(μM)^{2}}{μX μY} )^{2}  (1.3)

1For a metric probability space, λ_{1} is defined as the optimal constant in a Poincaré inequality; no Laplacian need exist.

for Neumann eigenvalues of compact Riemannian manifolds must have C_{1} ≥ 1/4. So, if we take μX and μY as “small” compared to μM, our inequality is optimal to first order. We also remark that we can fairly quickly see^{2} that the inequality in theorem 1.1 is stronger^{3} than that in equation 1.1 with C = 1 for all values of μX and μY^{4}.

Surprisingly, as indicated before, our result on Riemannian manifolds follows from our method of converting results from discrete math Laplacians to analysis Laplacians, along with the graph theoretic techniques of [CGY97]. Our inequality will hold for any Riemannian manifold and Laplacian as mentioned there. We shall also give a version of theorem 1.1 for higher eigenvalues.

In section 2 we will explain our method of converting results on Laplacians in discrete math to those in analysis. In section 3 we review the setup of [CGY97] and say what we mean by an “analytic” Laplacian. In section 4 we will use the method of section 2 and the techniques in [CGY97] to prove theorem 1.1; we prove several variants of this theorem there, including one for λ_{k} with k > 1. In section 5 we give a simple class of examples to show that we must have C_{1} ≥ 1/4 in an inequality 1.3.

2 The General Method

In this section we give a general method for converting results on discrete math Laplacians into those on analysis Laplacians. We will then describe how [FT] partially indicates this approach.

Our idea has two steps. First, if Δ is negative semidefinite Laplacian in analysis, then Δ̃ = 1 − cos \sqrt{−Δ} has two properties of a graph theoretic Laplacian:

(1) its spectrum^{5} lies in [0, 2], and

(2) the support of Δ̃u is within a distance of 1 to the support of u for any function u in L^{2}.

2Indeed, let A = μM/μX and B = μM/μY. It suffices to show that log(AB) exceeds cosh^{−1} \sqrt{(A − 1)(B − 1)}. Taking cosh of both sides reduces this to comparing (AB)/2 + 1/(2AB)

to \sqrt{(A − 1)(B − 1)}. Squaring and multiplying by 4 reduces this to showing (AB)^{2} + (AB)^{−2} + 4A + 4B − 4AB − 2 is positive. The gambit 4A + 4B ≥ 8\sqrt{AB} implies that it suffices to show that x^{2} + x^{−2} + 8\sqrt{x} − 4x − 2 is positive (for x ≥ 4). A computer investigation shows that y^{8} + 1 + 8y^{5} − 4y^{6} − 2y^{4} has no non-negative roots (where we substitute y^{2} = x); by hand one can also compute this polynomials Sturm sequence and apply the test between y = 0 and y = +∞ to see (by hand) that this polynomial has no positive roots.

3However, the result of Bobkov and Ledoux holds for any metric probability space, so our result is only stronger when an analytic Laplacian exists.

4Although Bobkov and Ledoux never mentioned it, it is possible to use the theorems in their paper to achieve any constant C_{1} > 1/4, with a C_{2} that depends on how close C_{1} is to 1/4, in equation 1.3.

5Graph theoretic and Markov chain Laplacians have their spectrum lying in [0, 2] if the Laplacians are appropriately normalized, as in [DK86, DK88, Chu93, CGY97, FT].

These properties, and possibly others, may make it possible to directly apply the same techniques used in discrete math to Δ̃.

The next step is to take the limit as the Riemannian metric is scaled^{6} by 1/ϵ and take ϵ → 0. This means that distances get scaled by 1/ϵ. This also means, for example, that we work with ϵ^{2}Δ instead of Δ, and with Δ̃ being 1 − cos(ϵ\sqrt{−Δ}).

Consider a situation where −Δ has eigenvalues 0 = λ_{0} < λ_{1} ≤ λ_{2} ≤ · · ·. We have to be careful about what we say about the eigenvalues of Δ̃^{ϵ} = 1 − cos(ϵ\sqrt{−Δ}). While it does have eigenvalues ν_{i} = 1 − cos(ϵ\sqrt{λ_{i}}) for each i, it is not true that ν_{1} is necessarily the second smallest eigenvalue. So if we want to use a technique involving the second smallest graph eigenvalue, we have to be careful to separate Δ̃^{ϵ} into its ν_{0} component, ν_{j} components for all ν_{j} ≥ ν_{1} and j ≥ 1, and finally the ν_{j} < ν_{1} and j ≥ 1. This third component will, in some sense, go to zero as ϵ → 0, and if things workout we will have an interesting theorem involving ν_{0} and ν_{1} ≈ ϵ^{2}λ_{1}/2.

We now explain how we came to the first step in our method. With any graph one traditionally associates a positive semidefinite Laplacian, Δ_{G}. In [FT] another type of positive semidefinite Laplacian is introduced, a so-called “edge-based” Laplacian, Δ_{E}, which much more closely resembles minus a Laplacian from analysis than does Δ_{G} (for example, the edge-based wave equation has wave propogation speed 1, where as that based on Δ_{G} has infinite speed). There are senses in which we may view Δ_{G} as similar to 1 − cos \sqrt{Δ_{E}}. This suggests that if Δ is a Laplacian from analysis, by forming Δ̃ = 1 − cos \sqrt{−Δ} we get a Laplacian that much more closely resembles a graph theoretic Laplacian.

3 Analytic Laplacians

In this section we explain what is meant by an “analytic Laplacian.” We begin by reviewing the setup in [CGY97]. We assume M is a metric space with Borel measure μ with μM < ∞. Let L be an operator on L^{2}(M, μ) (possibly unbounded) with for each s ∈ [0, ∞) a function P_{s}: spec(L) → ℝ with:

1. L is non-negative and self-adjoint,

2. Dom(L) is dense in L^{2}(M, μ) and contains the constants, and L maps the constants to 0, and

3. P_{s}(L) is bounded for each s with

supp(P_{s}(L)u) ⊂ {x ∈ M | dist(x, supp u) ≤ s}

6By scaling the metric by c we mean that in g = g_{ij} dx^{i} dx^{j} we multiply the g_{ij} by c^{2}.

for each u ∈ L^{2}(M, μ), i.e. P_{s}(L) enlarges the support^{7} of any function by a distance of at most s.

Definition 3.1 If the above conditions hold, we say that L is a standard Laplacian with propagation function P_{s}. We say that L is analytic if the above holds with P_{s}(λ) = cos(s\sqrt{λ}).

Many examples are given in [CGY97] of such M, μ, L with L an analytic Laplacian; they include

1. M is a complete Riemannian manifold (without boundary) of finite volume. L = −Δ with Δ the Laplace operator; more precisely, L is the unique self-adjoint extension of −Δ acting on C_{c}^{∞}(M).

2. M is a compact Riemannian manifold with boundary or a compact region of some other Riemannian manifold. L is as in item 1, imposing the Neumann boundary condition.

3. M is a complete Riemannian manifold, possibly of infinite volume. σ ∈ L^{1}(M, μ_{0}) is smooth, where μ_{0} is the Riemannian volume, such that with respect to μ = σμ_{0}, M is of finite volume. L is formed from −σ^{−1}∇ · (σ∇).

In [CGY97] it is remarked that if the greatest lower bound of the spectrum of L is λ_{0}, rather than 0, and λ_{0} has a corresponding positive eigenfunction, ψ, then ψ^{−1} ∘ L ∘ ψ − λ_{0} will be analytic. So a variant of theorems on analytic Laplacians hold for such Laplacians.

4 A Proof of the Theorem 1.1

We now assume that L is analytic, and set E to be the associated spectral measure, mapping Borel sets on ℝ to projections on L^{2}(M, μ). Let us assume that Spec L ⊂ {0} ∪ [λ_{1}, ∞) for some λ_{1} > 0, and that E({0}) is the projection onto the constants. If E({0}) is not the projection onto the constants, then λ_{1} = 0 and there is nothing to prove. Set

I = \{λ ∈ spec L | cos \sqrt{λ} > cos \sqrt{λ_{1}}\},

and let

π_{0} = E({0}), π_{−1} = E(I \ {0}), π_{1} = E(spec L \ I).

Δ_{G} defined to be 1 − cos \sqrt{L} has properties similar to a graph theory Laplacian, in that Δ_{G} is symmetric and positive semidefinite, Δ_{G} maps the constants to zero, Δ_{G} is bounded (namely ∥Δ_{G}∥ ≤ 2), and finally Δ_{G} enlarges supports of functions by at most a distance of 1. If also follows that if Q_{d} is any polynomial of degree ≤ d, then Q_{d}(Δ_{G}) expands supports by at most a distance of d. Now, if u, v ∈ L^{2}(M, μ) have supports whose distance between them is > d, then

7By the support of a function, u, in L^{2} we mean the support in the distribution sense, i.e. the complement of the union of all open sets Ω such that ∫ ϕu = 0 for, say, all bounded, continuous ϕ supported in Ω.

(Q_{d}(Δ_{G})u, v) = 0.

Let A_{i} = Q_{d}(Δ_{G}) ∘ π_{i} for i ∈ {−1, 0, 1}, and note that

(Q_{d}(Δ_{G})u, v) = (A_{0}u, π_{0}v) + (A_{1}u, π_{1}v) + (A_{−1}u, π_{−1}v).

Since

|(A_{0}u, π_{0}v)| = |Q_{d}(0)|∥π_{0}u∥_{2}∥π_{0}v∥_{2},

|(A_{1}u, π_{1}v)| ≤ ∥π_{1}u∥_{2}∥π_{1}v∥_{2} sup_{λ∈spec L\I}|Q_{d}(1 − cos \sqrt{λ})|,

|(A_{−1}u, π_{−1}v)| ≤ ∥π_{−1}u∥_{2}∥π_{−1}v∥_{2} sup_{λ∈I\\{0\}}|Q_{d}(1 − cos \sqrt{λ})|,

we obtain

∥π_{1}u∥_{2}∥π_{1}v∥_{2} [ sup_{λ∈spec L\I}|Q_{d}(1 − cos \sqrt{λ})|] ≥ ∥π_{0}u∥_{2}∥π_{0}v∥_{2}|Q_{d}(0)|−

∥π_{−1}u∥_{2}∥π_{−1}v∥_{2} [sup_{λ∈I}|Q_{d}(1 − cos \sqrt{λ})|]

Note that this inequality is just a variant of proposition 2.1 in [CGY97] (ū and u^{(1)} in [CGY97] correspond to our π_{0}u and π_{1}u + π_{−1}u). Now set T_{n} to be the standard Chebychev polynomial determined via

T_{n}(cos α) = cos(nα),

(we may replace cos by cosh if we like), and set^{8}

Q_{d}(x) = T_{d}(ℓ(x)) where ℓ(x) = \frac{2 + λ_{G,1} − 2x}{2 − λ_{G,1}}

where we set λ_{G,1} = 1 − cos \sqrt{λ_{1}} and we assume that d is an integer (compare this choice to the proof of theorem 4.3 in [CGY97]). Since ℓ(λ_{G,1}) = 1 and ℓ(2) = −1, Q_{d} maps [λ_{G,1}, 2] to [−1, 1]. Also

sup_{λ∈I}|Q_{d}(1 − cos \sqrt{λ})| ≤ Q_{d}(0) = T_{d} (\frac{2 + λ_{G,1}}{2 − λ_{G,1}} ) .

8The point of choosing Q_{d} like this is that the maximum over all polynomials P of degree d of \frac{P (0)}{max_{x∈[λ_{G,1},2]} |P (x)|} is attained for P = Q_{d}.

Hence we have

∥π_{1}u∥_{2}∥π_{1}v∥_{2} ≥ [∥π_{0}u∥_{2}∥π_{0}v∥_{2} − ∥π_{−1}u∥_{2}∥π_{−1}v∥_{2}] T_{d} (\frac{2 + λ_{G,1}}{2 − λ_{G,1}} ) .  (4.1)

Now let us fix u, v ∈ L^{2}(M, μ) and apply the same argument with Δ_{G}^{ϵ} = 1 − cos(ϵ\sqrt{L}) for small ϵ > 0; now Δ_{G}^{ϵ} extends supports by at most a distance ϵ. We conclude that we may use T_{d/ϵ} instead of T_{d} in the above argument and that we have^{9}

∥π_{1}^{ϵ}u∥_{2}∥π_{1}^{ϵ}v∥_{2} ≥ [∥π_{0}u∥_{2}∥π_{0}v∥_{2} − ∥π_{−1}^{ϵ}u∥_{2}∥π_{−1}^{ϵ}v∥_{2}] T_{d/ϵ} (\frac{2 + λ_{G,1}^{ϵ}}{2 − λ_{G,1}^{ϵ}} ) ,

where π_{±1}^{ϵ} is as before, based on

I^{ϵ} = \{λ ∈ spec L | cos(ϵ\sqrt{λ}) > cos(ϵ\sqrt{λ_{G,1}})\}

(and we are assuming that d/ϵ ∈ ℤ). Clearly

I^{ϵ} \ \{0\} ⊂ (spec L) ∩ [(2π − ϵ\sqrt{λ_{G,1}})^{2}/ϵ^{2}, ∞) ,

and so π_{−1}^{ϵ}u → 0 and similarly for v as ϵ → 0^{+}. Hence as ϵ → 0^{+} we also get π_{1}^{ϵ}u → u − π_{0}u and similarly for v. We also have that as ϵ → 0^{+}

λ_{G,1}^{ϵ} = 1 − cos(ϵ\sqrt{λ_{1}}) = ϵ^{2}λ_{1}/2 + O(ϵ^{4}),
\frac{2 + λ_{G,1}^{ϵ}}{2 − λ_{G,1}^{ϵ}} = 1 + ϵ^{2}λ_{1}/2 + O(ϵ^{4}),

cosh^{−1} (\frac{2 + λ_{G,1}^{ϵ}}{2 − λ_{G,1}^{ϵ}} ) = ϵ\sqrt{λ_{1}} + O(ϵ^{2}),

T_{d/ϵ} (\frac{2 + λ_{G,1}^{ϵ}}{2 − λ_{G,1}^{ϵ}} ) = cosh(d\sqrt{λ_{1}}) + O(ϵ).

So taking ϵ → 0^{+} with d/ϵ ∈ ℤ, and then taking d tending to the distance between the supports of u and v, we conclude the following.

Theorem 4.1 Let M, μ, L, λ_{1} be as before with L analysis-like. Then for any u, v ∈ L^{2}(M, μ) whose supports are at a distance d, we have

cosh(d\sqrt{λ_{1}}) ≤ \frac{∥u − π_{0}u∥_{2}∥v − π_{0}v∥_{2}}{∥π_{0}u∥_{2}∥π_{0}v∥_{2}} .

the Laplacian by ϵ^{2}, etc.

9Another way to view this approach is to think of the case in Riemannian geometry, and to scale the metric by 1/ϵ (i.e. in g = g_{ij} dx^{i} dx^{j} we scale the g_{ij} by 1/ϵ^{2}). This scales distances by 1/ϵ, scales

Taking u and v to be the characteristic functions of two disjoint sets, X, Y, we conclude

d\sqrt{λ_{1}} ≤ cosh^{−1} \sqrt{\frac{μX^{c} μY^{c}}{μX μY}} ,

where d = dist(X, Y). This proves theorem 1.1.

□

The following theorem mildly generalizes the case where L is analytic; we do not know if it has any interesting applications; its proof is the same as the above.

Theorem 4.2 Let M, μ, λ_{1}, L be as above, with L a standard Laplacian. Let P_{s} be a propagation function such that (1) there exist constants C, α such that

P_{ϵ}(λ) = Cϵ^{2}λ^{α} + O(ϵ^{3})

for any λ, and (2)

P_{ϵ}(spec L) ⊂ [0, B_{ϵ}],

with

lim inf_{ϵ→0} B_{ϵ} = B > 0.

Then the preceeding theorem and corollary hold with \sqrt{λ_{1}} replaced by (4Cλ_{1}^{α}/B)^{1/2}.

As a silly example of this theorem, we may take L = (−Δ)^{k} where Δ is a Riemannian Laplacian and k a positive integer, and set P_{s}(λ) = 1 − cos[sλ^{1/k}] (and thereby recover the k = 1 result).

Finally we remark that the generalization of [CGY97] to the (k + 1)-st eigenvalue, λ_{k}(L), of L, holds as well. More precisely, let k ≥ 1 be an integer, and let λ_{k} be such that E([0, λ_{k})) is a projection onto a subspace of dimension k.

Theorem 4.3 Let u_{0}, …, u_{k} be k + 1 functions in L^{2}(M, μ) with supports which are at distance ≥ d of each other. Then

cosh(d\sqrt{λ_{k}}) ≤ max_{i≠j} \frac{∥u_{i} − π_{0}u_{i}∥_{2}∥u_{j} − π_{0}u_{j}∥_{2}}{∥π_{0}u_{i}∥_{2}∥π_{0}u_{j}∥_{2}} .

Proof We simply add the ideas of the proof of proposition 2.2 in [CGY97] into what we have already done. First of all, we may assume that (π_{0}u_{i}, π_{0}u_{j}) ≥ 0 for each i, j, by having π_{0}u_{i} be a non-negative constant for each i, replacing u_{i} by −u_{i} if necessary.

Let I and Q_{d} be as before, with λ_{k} replacing λ_{1}, and set

π̃_{0} = E([0, λ_{k})), π_{−1} = E(I \ [0, λ_{k})), π_{1} = E(spec L \ I).

Since the supports of u_{i} and u_{j} are at distance > d for i ≠ j we obtain

(Q_{d}(Δ_{G})u_{i}, u_{j}) = 0

and therefore

−(Q_{d}(Δ_{G})π_{1}u_{i}, π_{1}u_{j}) = (Q_{d}(Δ_{G})π̃_{0}u_{i}, π̃_{0}u_{j}) + (Q_{d}(Δ_{G})π_{−1}u_{i}, π_{−1}u_{j})

Since

∥π_{1}u∥_{2}∥π_{1}v∥_{2} ≥ |(Q_{d}(Δ_{G})π_{1}u_{i}, π_{1}u_{j})|
≥ |(Q_{d}(Δ_{G})π̃_{0}u_{i}, π̃_{0}u_{j})| − |(Q_{d}(Δ_{G})π_{−1}u_{i}, π_{−1}u_{j})|
≥ |(Q_{d}(Δ_{G})π̃_{0}u_{i}, π̃_{0}u_{j})| − sup_{λ∈I}|Q_{d}(1 − cos \sqrt{λ})|∥π_{−1}u_{i}∥_{2}∥π_{−1}u_{j}∥_{2}

Let π_{0} be the projection onto the constants, and set

π_{new} = π̃_{0} − π_{0}.

Then π_{new} is a projection onto a subspace of dimension k − 1, and defining

⟨u, v⟩ = (Q_{d}(Δ_{G})π_{new}u, π_{new}v)

gives a positive definite inner product, provided that Q_{d} maps [0, λ_{G,k}) to the positive reals. Our choice of Q_{d} (with λ_{G,k} = 1 − cos \sqrt{λ_{k}} replacing λ_{G,1}) clearly does so.

Since for any k + 1 vectors in a k − 1 dimensional Euclidean space there are two with a non-negative inner product (see [CGY96]), there are i, j with ⟨u_{i}, u_{j}⟩ ≥ 0. For this i, j we have

(Q_{d}(Δ_{G})π̃_{0}(u_{i}), π̃_{0}(u_{j})) = (Q_{d}(0)π_{0}(u_{i}), π_{0}(u_{j})) + ⟨u_{i}, u_{j}⟩
≥ (Q_{d}(0)π_{0}(u_{i}), π_{0}(u_{j})).

For this i, j we have:

∥π_{1}u_{i}∥_{2}∥π_{1}u_{j}∥_{2} ≥ [∥π_{0}u_{i}∥_{2}∥π_{0}u_{j}∥_{2} − ∥π_{−1}u_{i}∥_{2}∥π_{−1}u_{j}∥_{2}]T_{d} (\frac{2 + λ_{G,k}}{2 − λ_{G,k}} ) .

Repeating this argument with Δ_{G}^{ϵ} = 1 − cos(ϵ\sqrt{L}) as before yields the theorem.

Corollary 4.4 Let X_{0}, …, X_{k} be k + 1 disjoint Borel subsets of M, whose distance between any two is at least d, and with other notation as before. Then

λ_{k} ≤ \frac{1}{d^{2}} max_{i≠j} (cosh^{−1} \sqrt{\frac{μX_{i}^{c} μX_{j}^{c}}{μX_{i} μX_{j}}} )^{2} .

5 An Example Extremal to First Order

In this section we give a simple and more or less well-known example to show that the optimal constant, C_{1}, in the inequality 1.3 is indeed 1/4, as long as the inequality holds for the Neumann eigenvalues of manifolds with boundaries. If the inequality also holds for metric probability spaces, then C_{1} ≥ 1/4 can be easily seen from the standard “exponential distribution” (studied in, for example, [BL97]); this space is just the nonnegative reals with the usual metric, endowed with the probability measure e^{−x} dx. To emulate this in a Riemannian manifold, we take the surface of revolution generated by y = e^{−x} for x in an interval, [L, R], with L and R − L large^{10}; these and related surfaces are well known examples of interesting spectral phenomena— see [Bro84] for example^{11}. We will see that in our case we must take L → ∞ with (R − L) → ∞ to conclude that C_{1} ≥ 1/4.

So we consider for reals L < R the surface of revolution of y = e^{−x} for x between L and R,

M_{L,R} = {(x, y, z) ∈ ℝ^{3} | y^{2} + z^{2} = e^{−2x}, x ∈ [L, R]}.

We shall prove the following.

Lemma 5.1 For λ_{1} being first non-zero Neumann eigenvalue of M_{L,R} we have

λ_{1}(M_{L,R}) ≥ \frac{1}{4(1 + e^{−2L})}.

Lemma 5.2 Let X, Y be the subsets of M_{L,R} consisting of those points (x, y, z) with x ∈ [L, L + 1] and, respectively, x ∈ [R − 1, R]. Then dist(X, Y) ≥ R − L − 2 and

log \frac{(μM)^{2}}{μX μY} ≤ R − L + 1 − 2 log(e − 1) + log(1 + e^{−2L}).

An immediate consequence of these two lemmas is the following theorem.

Theorem 5.3 Let equation 1.3 hold for all compact manifolds with boundary with the Neumann condition. Then C_{1} ≥ 1/4.

Since C_{2} can never be less than 1/4 (or else take μX = μY to be nearly (μM)/2), our lemmas give no improvement over this trivial bound on C_{2} for the case C_{1} = 1/4.

We remark that if equation 1.3 holds only for compact manifolds without boundary, then it seems probable to the authors that one can add “caps” or “bulbs” (as in [Bro84]) to the x = L and x = R ends and get a similar theorem.

10Richard Froese has pointed out that we could simplify this section by modifying our Riemannian metric. Namely, we could take ds^{2} = dt^{2}+e^{−2t} dθ^{2} for the metric (see below), and then all the 1+e^{−2t} and 1 + e^{−2L} factors would become 1.

11We remark that [Bro84] is mentioned in [CGY97] as evidence that their inequality might be improvable by roughly a factor of 1/4.

To prove the lemmas we recall some standard formulas in Riemannian geometry. First, if a Riemannian manifold has local coordiates x^{1}, …, x^{n} and Riemannian metric g = g_{ij} dx^{i} dx^{j}, then recall (using the standard summation convention and the convention that a_{,j} denotes ∂a/∂x^{j})

dV_{g} = \sqrt{det g} dx^{1} ⋯ dx^{n}
∇u = g^{ij}u_{,j}(∂/∂x^{i})
|∇u|^{2} = g^{ij}u_{,i}u_{,j}
Δu = (det g)^{−1/2}(\sqrt{det g} g^{ij}u_{,i})_{,j}

where dV_{g} is the Riemannian volume and u a sufficiently smooth function. Placing coordinates t ∈ [L, R] and θ ∈ ℝ/(2πℤ) on M_{L,R} via

x = t, y = e^{−t} cos θ, z = e^{−t} sin θ,

the metric induced from ℝ^{3} to M_{L,R} is

ds^{2} = dx^{2} + dy^{2} + dz^{2} = (1 + e^{−2t}) dt^{2} + e^{−2t} dθ^{2}.

Plugging in g_{11} = 1 + e^{−2t}, g_{12} = g_{21} = 0, and g_{22} = e^{−2t} into the above formulas yields:

dV_{g} = e^{−t}\sqrt{1 + e^{−2t}} dt dθ
|∇u|^{2} = (1 + e^{−2t})^{−1}u_{t}^{2} + e^{2t}u_{θ}^{2}
Δu = \frac{1}{e^{−t}\sqrt{1 + e^{−2t}}} [(\frac{e^{−t}u_{t}}{\sqrt{1 + e^{−2t}}} )_{t} + (\frac{\sqrt{1 + e^{−2t}}}{e^{−t}} ) u_{θθ}] .

We finish by proving the lemmas. The first statement of lemma 5.2 just follows from the fact that X and Y are separated by R − L − 2 in their x coordinate alone. For the second statement, we have

μX = ∫_{L}^{L+1} 2πe^{−t}\sqrt{1 + e^{−2t}} dt ≥ ∫_{L}^{L+1} 2πe^{−t} dt = 2πe^{−L−1}(e − 1)

and similary for μY. Combining this with

μM = ∫_{L}^{R} 2πe^{−t}\sqrt{1 + e^{−2t}} dt ≤ 2π\sqrt{1 + e^{−2L}} ∫_{L}^{∞} e^{−t} dt

yields the lemma.

Lemma 5.1 requires more computation. Let f be the eigenfunction corresponding to λ_{1}. We shall prove lemma 5.1 by first assuming that f = f(t) is independent of θ. Then we will show that this assumption is justified.

So assume f = f(t). Then f, as a function of t alone, is the second Neumann eigenfunction for the Rayleigh quotient

R(u) = \frac{∫ |∇u|^{2} dV_{g}}{∫ u^{2} dV_{g}}

= \frac{∫_{L}^{R} u_{t}^{2}(e^{−t}/\sqrt{1 + e^{−2t}}) dt}{∫_{L}^{R} u^{2}e^{−t}\sqrt{1 + e^{−2t}} dt}.

Hence

R(u) ≥ (1 + e^{−2L})^{−1}R̃(u),

where

R̃(u) = \frac{∫_{L}^{R} u_{t}^{2}e^{−t} dt}{∫_{L}^{R} u^{2}e^{−t} dt}.

It follows from the min-max principle that λ_{1} is at least as big as (1 + e^{−2L})^{−1} times the second Neumann eigenvalue for R̃; this later eigenvalue, λ, has corresponding eigenfunction u with

(e^{−t}u_{t})_{t} = −λue^{−t},

i.e.

u_{tt} − u_{t} + λu = 0.

The general solution to the last equation is

u(t) = A_{1}e^{r_{1}t} + A_{2}e^{r_{2}t}, where r_{1,2} = \frac{1 ± \sqrt{1 − 4λ}}{2}

provided that λ ≠ 1/4. The Neumann conditions at t = L and t = R are immediately seen to imply that (R − L)(r_{2} − r_{1}) ∈ 2πiℤ, and in particular λ > 1/4. Hence either λ = 1/4 or λ > 1/4 for the second Neumann eigenvalue. It remains to show that f = f(t).

To show that f = f(t), assume to the contrary that f depends on θ. We will use separation of variables to show that λ_{1} would then be larger than 1/4 (assuming, say, L ≥ 0). Indeed, solutions to Δu = −λu satisfy

(h_{1}u_{t})_{t} + h_{2}u_{θθ} = −λh_{3}u,

where the h_{i} = h_{i}(t) are given by

h_{1} = \frac{e^{−t}}{\sqrt{1 + e^{−2t}}} , h_{2} = \frac{\sqrt{1 + e^{−2t}}}{e^{−t}} , and h_{3} = e^{−t}\sqrt{1 + e^{−2t}}.

If u has the form A(t)B(θ), then we have

B′′/B = [−λh_{3} − (h_{1}A′)′/A]/h_{2}

which must be constant, since the left- and right-hand-sides respectively depend only on t and θ, respectively. Hence we can take B(θ) = e^{inθ} with n ∈ ℤ, with corresponding A’s satisfying

−(h_{1}A′)′ + n^{2}h_{2}A = λh_{3}.  (5.1)

Standard ODE theory shows that for fixed n ∈ ℤ the set of such A with Neumann condition span L^{2}[L, R], and so such solutions u(t, θ) = A(t)e^{inθ} give a basis for L^{2}(M_{L,R}). Hence if f depends on θ, then λ_{1} = λ for some Neumann solution to equation 5.1 with n ≠ 0. But such a λ equals the Rayleigh quotient R_{n}(A) with

R_{n}(A) = \frac{∫_{L}^{R}[h_{1}(A′)^{2} + n^{2}A^{2}h_{2}] dt}{∫_{L}^{R}[A^{2}h_{3}] dt}.

Ignoring the clearly non-negative h_{1}(A′)^{2} term in the numerator, we see that

R_{n}(A) ≥ \frac{∫_{L}^{R}[n^{2}A^{2}h_{2}] dt}{∫_{L}^{R}[A^{2}h_{3}] dt} ≤ n^{2} min(h_{2}/h_{3}) = n^{2}e^{2L}.

It then follows that

λ_{1} ≥ n^{2}e^{2L} ≥ e^{2L},

which certainly exceeds 1/4 when L ≥ 0.

References

[AM85] N. Alon and V. D. Milman. λ_{1}, isoperimetric inequalities for graphs, and superconcentrators. J. Combin. Theory Ser. B, 38(1):73–88, 1985.

[BL97] S. Bobkov and M. Ledoux. Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution. Probab. Theory Related Fields, 107(3):383–400, 1997.

[Bro84] Robert Brooks. On the spectrum of noncompact manifolds with finite volume. Math. Z., 187(3):425–432, 1984.

[CFM94] F. R. K. Chung, V. Faber, and Thomas A. Manteuffel. An upper bound on the diameter of a graph from eigenvalues associated with its Laplacian. SIAM J. Discrete Math., 7(3):443–457, 1994.

[CGY96] F. R. K. Chung, A. Grigor’yan, and S.-T. Yau. Upper bounds for eigenvalues of the discrete and continuous Laplace operators. Adv. Math., 117(2):165– 178, 1996.

[CGY97] F. R. K. Chung, A. Grigor’yan, and S.-T. Yau. Eigenvalues and diameters for manifolds and graphs. In Tsing Hua lectures on geometry & analysis (Hsinchu, 1990–1991), pages 79–105. Internat. Press, Cambridge, MA, 1997.

[Chu93] F.R.K. Chung. Laplacians of graphs and Cheeger inequalities. In Combinatorics Paul Erdös is eighty, pages 1–16. Bolyai society mathematical studies, 1993.

[DK86] J. Dodziuk and W.S. Kendall. Combinatorial Laplacians and isoperimetric inequality. From local times to global geometry, control and physics, 1986.

[DK88] Jozef Dodziuk and Leon Karp. Spectral and function theory for combinatorial Laplacians. Geometry of Random Motion, Contemp. Math. 73, pages 25–40, 1988.

[FT] Joel Friedman and Jean-Pierre Tillich. Wave equations for graphs and the edge based Laplacian. to appear.

[LPS88] A. Lubotzky, R. Phillips, and P. Sarnak. Ramanujan graphs. Combinatorica, 8(3):261–277, 1988.

[Moh91] Bojan Mohar. Eigenvalues, diameter, and mean distance in graphs. Graphs Combin., 7(1):53–64, 1991.

[MS86] Vitali D. Milman and Gideon Schechtman. Asymptotic theory of finitedimensional normed spaces. Springer-Verlag, Berlin, 1986. With an appendix by M. Gromov.

[Sar90] P. Sarnak. Some applications of modular forms, volume 99 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1990.