Tubes and eigenvalues for negatively curved manifolds

The Journal of Geometric Analysis Volume 3, Number 1, 1993

Tubes and Eigenvalues for Negatively Curved Manifolds

By P. Buser, B. Colbois, and J. Dodziuk

ABSTRACT. We investigate the structure of the spectrum near zero for the Laplace operator on a complete negatively curved Riemannian manifold M. If the manifold is compact and its sectional curvatures K satisfy 1 < K < 0, we show that the smallest positive eigenvalue of the Laplacian is bounded below by a constant depending only on the volume of M. Our result for a complete manifold of finite volume with sectional curvatures pinched between - a 2 and - 1 asserts that the number of eigenvalues of the Laplacian between 0 and (n - 1 )2/4 is bounded by a constant multiple of the volume of the manifold with the constant depending on a and the dimension only.

1. Introduction

We are considering the spectrum of the Laplacian on an n-dimensional Riemannian manifold M of finite volume and strictly negative curvature. The curvature is normalized in such a way that either the upper or the lower sectional curvature bound is - 1. In either case, an eigenvalue of the Laplacian on M is called small if it is contained in the interval (0, (n - 1)2/4). The number (n - 1)2/4 is not arbitrary. It is, for instance, the bottom of the spectrum of the n-dimensional hyperbolic space and is also the "critical" value in Selberg's trace formula. The role of the small eigenvalues was observed first by Huber and Selberg in connection with hyperbolic lattice point problems and with the distribution of lengths of the closed geodesics on a compact Riemarm surface [24, 25, 26, 36]. The small eigenvalues are also of great interest for locally symmetric

spaces of rank one [5, 17, 22]. We remark that if M is noncompact and has upper curvature bound - 1 , then (n - 1)2//4 is a lower bound of the essential spectrum.

It was shown in [35] and [11] that if the sectional curvature K has the bounds - 1 _< K _< - k 2, then the first eigenvalue A1 (M) is bounded below by c (n ) ( k /V) 2 where V is the volume of M and c(n) is a constant that depends only on the dimension. This bound goes to zero as k approaches zero. We prove here a lower bound that is independent of k. This result holds if M is compact and has dimension n >_ 4 (Theorem 3.1). In the noncompact case and in dimension two and three there are counterexamples (Example 4.4). For the proof of Theorem 3.1 we use methods of [14] and [11] together with a result of Gromov [20] about the volume of tubes in

negatively curved manifolds.

Math Subject Classification Primary 58G25; Secondary 53C21. Key Words and Phrases Cusps, eigenvalues, Laplacian, negative curvature, tubes. Research supported in part by the Swiss National Science Foundation, the US National Science Foundation, and

the PSC-CUNY Research Award Program.

~)1993 CRC Press, Inc. ISSN 1050-6926

2 e. Buser, B. Colbois, and J. Dodziuk

It was shown in [6] that for a compact hyperbolic threefold, the number of eigenvalues occurring in the interval (0, 1) is bounded above by cV, where c is a constant. We extend this result to any n-dimensional negatively curved manifold of finite volume and with sectional curvature bounds - a 2 _ K _< - 1 (Theorem 3.6). Here the constant c depends on n and on

the lower curvature bound. We give an example (Example 4.3) showing that the lower curvature bound is necessary.

The approach to Theorem 3.6 is through a study of the geometry of tubes and cusps which is carried out in Section 2. By definition, tubes and cusps are the bounded and unbounded components

of the thin part of the manifold respectively. The proof in [6] relied in an essential way on the

fact that in hyperbolic threefolds, tubes are round, i.e., the distance from the core geodesic to the boundary of the tube is constant. In the general case this is no longer true (in higher dimensions this fails even if M has constant curvature). Furthermore, the boundaries of tubes and cusps are not necessarily smooth everywhere. This is due to the somewhat ambiguous definition of what is to be the "thin part of M."

Our main goal in Section 2 is to modify the tubes (and cusps) by constructing a new boundary near the old one that has controlled intrinsic injectivity radius and controlled intrinsic curvature

bounds. The main result of this section (Theorem 2.14) could be obtained by the technique

of Cheeger and Gromov [9]. We give an independent proof making use of the knowledge of

geometry of tubes and cusps. The new tubes and cusps admit a decomposition into a controlled number of pieces ("pieces of cheese") whose Cheeger constant is bounded below by n - 1. The somewhat difficult proof of the estimate of the Cheeger constant is the same as in [6] and will

not be repeated here. Instead we show how to estimate the first positive Neumann eigenvalue of a "piece of cheese" directly, without using isoperimetric constants, if the ambient manifold is a locally symmetric space of rank one. This proof is given at the end of Section 3 (Lemma 3.12). In Section 4 we construct a number of examples and counterexamples.

2. T u b e s a n d c u s p s

In this section we establish the terminology and notation, review known facts about the "thick and thin" decomposition of negatively curved manifolds, and prove some new results about the boundaries of tubes and cusps (Theorem 2.14 and Lemma 2.7). [20] and [2] are good references for most of the material in this section.

We consider complete manifolds M of finite volume, of n > 3 dimensions, with sectional curvatures K satisfying

- a 2 < K _< - 1 , (2.1)

for a constant a. 2k/with a covering group 1 ~ is a fixed universal covering with the projection 7r : M ) M . Margulis' Lemma [2] states that there exists a constant

/2 = a - - lCn , (2.2)


where cn depends only on the dimension such that the following is true. Let p E M be any point and let o~ and/3 be two geodesic loops at p. If both l (a ) and I(/3) [ l (a) denotes the length of a] are smaller than or equal to 2# then a and/3 generate an almost nilpotent subgroup of 7rl (M, p). The constant/z is referred to as the Margulis constant.

A geometric consequence of Margulis' Lemma is the decomposition of M into its "thick" and "thin" parts. The precise definition of "thick" and "thin" is usually adapted to momentary needs. Roughly speaking, a point x E M belongs to the thick part if the injectivity radius L(x) is greater than or equal to # and otherwise belongs to the thin part. Denote by M u the set

{ x E M [ L(x) < /z }. The first consequence of Margulis' Lemma is that M \ M u 7s @. M u may be empty but, if it is not, it has a very nice geometric structure that also follows from Margulis' Lemma. Namely, Mu is a union of finitely many components whose structure we now describe. A component of Mu is called a tube if it is relatively compact and a cusp otherwise. We begin by describing the geometry of tubes. Every tube T contains a unique closed geodesic 7 of length/(7) < 2/z. Later on we shall restrict our attention to geodesics whose length is very small compared to #. For every x E 7 and every tangent vector v at x perpendicular to 7, we consider a unit speed geodesic ray 6 emanating from x in the direction of v. In every interval [0,/3] such that L(6(s)) <_ i ~ for all s E [0,/3], the function s ~-+ c(6(s)) is strictly increasing. Moreover, there exists R, depending on x and v such that L(6(R)) = # and ~(6(s)) < /z if 0 < s < R.

The arc 6([0, R]) will be called a maximal radial arc or simply a radial arc. Different radial arcs are disjoint, except possibly for their initial points, and T is the union of all radial arcs emanating from '7- In particular, T is homeomorphic to 7 x B n- l , where B n-x is the open unit ball in R n- l , since M is assumed oriented for convenience. We denote by 6x the radial arc determined

b y x E T r \ 7 .

Alternatively, tubes can be described in terms of the action of the deck group F on M.

Let A-y be the isometry of )~f corresponding to the class of "7 in 7rl (M) = F. Let Tq be the component of 7r-l(T.~) containing the axis of A~. Then Tq is precisely invariant under the cyclic group (A~). Therefore the injectivity radius of x = 7r(~), ~ E T~ is given by

~(x) = lmink~z d(k, Akr~), Z

(2.3)

where d(~, ~) denotes the distance between ~ and ~.

We remark that the boundary of T~ need not be smooth in general and that maximal radial arcs in T~ have considerably varying lengths. This may happen even when the sectional curvature of M is constant. One reason for this is that the integer realizing the minimum in (2.3) may be different for different points of OT~. However, when the sectional curvature is constant and n = 3, the tubes are round, i.e., all radial arcs have the same length [6].

The structure of cusps is very similar. Intuitively, the geodesic at the core shrinks to a point that moves out to infinity. Thus a cusp is a union of disjoint maximal radial arcs which have infinite lengths and emanate from a common point at infinity. We make this more precise. Let C be a cusp, i.e., an unbounded component of M , . Choose a component C' of the inverse image 7r -1 (G'). It

is known [16] that r d = { o~ E F I a C = C } is equal to Fz = { a E r I a z = z } for some


Z E A: / (~) (the sphere at infinity of AS/) and that C is precisely invariant under l~z. Moreover, I'z preserves horospheres in M centered at z and acts on each of them with a compact quotient since C = l~z\C C M has finite volume. Consider a unit speed geodesic ~ : (-cx>, c~) ) M with the property, l ims__~ 6(s) = z, and its image 6 in M. Then lims._._~ ~(~(s)) = 0 and L(6(s)) is strictly increasing for s E ( -co , /3 ) if ~(~(s)) < a for s in this interval. The parameterization of 6 is unique only up to translation, but, once a parameterization is chosen (for example so that 6(0) lies on a fixed horosphere), we can define R as before to be the smallest value of s for which L(6(s)) = ~. The geodesic half-line 6 ( ( - c c , R]) will be called a maximal radial arc. Every cusp is a union of disjoint maximal radial arcs. As above, we emphasize that the boundary of a cusp need not be smooth and cusps are not necessarily round, i.e., the boundary of a cusp need not lie in the image of a horosphere.

We need to obtain some information about the geometry of the boundary of a cusp or a tube. In case of tubes, there is a preliminary result, undoubtedly well known, which we could not locate in the literature.

L e m m a 2.4. Let T.~ be a tube around a geodesic 7 with I(7 ) < Iz. Then the length R of every radial arc satisfies

cn <_ l(7)a -(n-l) sinhn-l(aR),

where cn, #, and a are as in (2.1).

Proof. Note first, that for x, y E M I~(x) - L(y) l < d(x, y). If x E % then ~(x) < #/2. Let 6 : [0, R] ) M be a maximal radial arc. Choose a point y E 6 with injectivity radius 3/z/4. Then d(y, 6(R)) _> #/4. In addition, if z e M and d(y, z) < u / a , then L(z) < #. It follows that the ball Bu/4(y ) is contained in T-y. Now we do a simple volume comparison. On the one hand,

On the other,

B~,/,,(y) C_ { z e T. t I d(z, 3') _< R }.

The volume of this set is bounded from above by the volume of the tubular neighborhood of radius R of a geodesic segment of length l(7 ) in the simply connected space of constant curvature - a 2. This yields the desired inequality. [ ]


We see from the proof of the lemma that i f / ( 7 ) is sufficiently small then points of Tr whose injectivity radius is equal to # / 2 are at large distances from 7. More precisely, there exist constants cl, c2, depending only on n, such that, if

/(7) -< cle-~='~l-zna'~-I (2.5)

then d(x, 7) > 10 for every point x E T./with c(x) = # /2 .

We now give our definition of thick and thin decomposition. The only tubes that contribute to the thin part, M~n, are those that surround geodesics satisfying the inequality (2.5). M~n consists of points that belong to such tubes or to cusps and have injectivity radius at most # /2 . Further, Mt~r = M \ Mtran. Thus OMt~n consists of points x with injectivity radius ~(x) = # / 2 but

1 ca n ,~ i (2.6)

which may be considerably smaller than # /2 . From now on we shall refer to the components of M~n as tubes and cusps and use the notation

U.~ = { x E T.~ [L(x) < # / 2 } .

We want to replace every component of Mt~ by a "tube" or a "cusp" with smooth boundary H , which is a small perturbation of the original boundary and so that H has controlled sectional curvature. In addition we require H to satisfy a certain transversality condition with respect to the radial arcs. The first step is the lemma below, which we state and prove for the

case of a tube.

L e m m a 2.7. Let x, y E OU r and d(x, y) <_/z/10. Let t9 be the angle between ~ and the geodesic segment with endpoints x and y as in Figure 1. Then

&<O<u-& 20 - 20

Proof. We prove the first inequality. The second one foUows by reversing the roles of x and y. Let y ' E 6 u be the point such that d(7, x) = d(% y') . To fix attention, we assume that y is farther from 7 than x. Let u = d(x, y'). The arc u is almost orthogonal to 6u, as follows from Lemma 2.13 below. Therefore it suffices to prove an upper bound for the ratio v / u (see Figure 2). Note that L(y') > ~(x) - u = # / 2 - u. On the other hand ~(y) = # / 2 so that

~(y) < /z (2.8) e(y') - # - 2u"

6 p. Buser, B. Colbois, and J. Dodziuk

,y

x

X

Figure 1.

Figure 2.

We also claim that

L(y___)_) > eV/2" (2 .9 ) , ( y , ) -

Assume this for a moment. Then (2.8) and (2.9) yield

- # - 2 u '

which implies that v /u < 5 / # since u < #/10. []

To complete the proof of Lemma 2.7 we need to establish the claim (2.9). To do this, it is convenient to work in the universal coveting and use the characterization of injectivity radius given in (2.3). Let ~, ~t be inverse images of y, y ' respectively in T~ lying on a common perpendicular 6 0 to "~. Let l be an exponent such that d(~, A~) = / ~ and d(~, A ~ ) _> # for all m 5~ 0. Then

L(y) d(~,A~l) ~(y"---~ >- d(fl',A~l')"

Since # > ~(y') > 4_z and d(~', @) > 9 the claim will follow from the lemma below, which is actually its infinitesimal version.


Figure 3.

L e m m a 2.10. Let y, z E 2~/I be two points such that d(y, ~) = d(z, ~) > 9 and 2p > d(y, z) > 8u Parameterize the radial arcs 6 v and 6z by the distance from ;f. If l (s) =

~ 5 "

d(6v(s ), 6z(s)) then, for s > 9

dl l

ds - 2

Proof. Note first that by the formula for first variation of arc length d l /ds = cos a + c o s / 3 (see Figure 3). Therefore we want to estimate o~ and/3. By [8, Example 6.5],

c o s h c - 1 < cosh 1 - 1

cosh 2 r "

Since 1 and c are very small and r is large, we see that

c < 2 le - ' , (2.11)

where we ignored a multiplicative constant that is very close to one. Consider a geodesic segment h connecting 6v(0 ) with z and geodesic triangles r lh and chr. Let rlh and chr be the triangles in the hyperbolic space (curvature - 1 ) with sides of lengths r, l, h and c, h, r respectively. Denote by czt,/31 ~, t~l, r the angles of rlh and chr corresponding to c~,/3', n, r respectively. If ~b is any of these angles then, by [8, Corollary 6.4.3], ~b < r It follows then from the law of sines that

I~ 1 ~ 2le-" < 2le -9 and r _< ce-9 <_ 2le-18. (2.12)

The inequalities (2.11) and (2.12) will mean for us that c and the angles ~, r and r are negligible. Thus, we pretend that the configuration in Figure 3 is a triangle rhl and r = h. The angle c~ can now be determined from the law of cosines

C O S O~ 1 " -"

c o s h r ( c o s h l - 1 ) > l

sinh r sinh l - 2

Again we ignored the factor that is very close to one. Since ot < c~l, this proves that cos a > l /2. By symmetry the same holds for/3. Since we repeatedly ignored higher order terms, etc., in this

8 P. Buser, B. Colbois, and J. Dodziuk

6x

Figure 4.

argument we can only claim that d l /d s = cos c~ + cos/3 > I/2. A more careful argument would yield the lower bound of the form f l with f very close to 1. [ ]

For future use we need another angle estimate.

Lemma 2.13. Suppose ;y is a geodesic in M and z E ](/f is a point such that d(z , '~) >_ 9.

Let 6x be the perpendicular to ~/ from z, and 7" a geodesic ray emanating from x perpendicularly to 6z. Choose a point y on "r such that l = d(z , y) <_ #. Then the angle/3 between "r and 6 u (see Figure 4) satisfies

~/2 >/3 > ~/2 - 21a.

Proof . The first inequality is obvious and the angles r and tc can be estimated as above by a comparison with curvature - 1 model.

r --< r --< 2/e -9, tc ~_~ /~1 ~--_ 21e -9

Now let c~a,/3a', t% be the angles, in the triangle r~lh with sides r~ = d(x, ;/), l, h in the plane of constant curvature - a z, corresponding to c~ = 7r/2,/3' and tc respectively. As above, we see that

/3o' < /3' , t% < t~ < tq < 21e -9.

The Gauss-Bonnet formula applied to the triangle r~lh yields

~ / 2 - ~ : + ~ / 2 - / 3 : ' = <~area(~-;7-~) + ,~o.


Using the estimate of Cheeger's constant in [6, Lemma 1.11] aArea < l, we obtain

7r/2 - ~a t ~ aZ .dr_ 2 le -9

so that

>_ 13' -- r >_ lr/2 - 2al. []

R e m a r k , Lemmas 2.7 and 2.13 have obvious analogs for cusps. The proofs in this case are in fact somewhat easier and we omit them. [ ]

We now state the main result of this section.

T h e o r e m 2.14. Let 7 be a geodesic in M satisfying (2.5). There exists a smooth hypersurface H contained in T r \ 7 with the following properties:

(i) The angle O between the radial vector 7-~ and the exterior normal to H is less than 7r/2 - olfor a constant o~ = a ( a , n) E (0, 7r/2).

(ii) The sectional curvatures of H with respect to the induced metric are bounded in absolute value by a constant depending only on a and n.

(iii) Because of(i), H is homeomorphic to OU r by pushing along radial arcs. The distance between x E H and its image ~ E OU r satisfies d(x, "~) < #/50.

Proof . As a preliminary, we recall some facts about the geometry of distance tubes [18, w Let "~ be a geodesic in M projecting onto 3' and let ,_q, be the set of points of M at distance r from "~. Then the principal curvatures ~i of S . with respect to 7~ satisfy the following inequalities:

- atanhar <_ no <_ - tanhr and - acothar <_ t~i <_ - cothr (2.15)

for i = 1 , 2 , . . . , n --2. It follows that the sectional curvatures of S . stay bounded if r is bounded away from zero.

We now proceed to define H as the zero set of a function defined in a neighborhood of OU r. Whenever convenient we shall work in the universal covering without explicitly mentioning it.

Thus, we do all our constructions in U r or, equivalently, in Ur = { x E Tr [ 7r(x) E U r } equivariantly with respect to the cyclic group (A.r). Choose a maximal set of points {qi} in OU r at distances d(qi, qj) >_ e for i ~ j . e is a small parameter (much smaller than #) whose value will be fixed later. The balls B~/2 (qi) are disjoint and B , (qi) cover OU r. Choose a smooth bump function r [0, oo) ~ [0, 1] such that r = 1 on [0, 1] and r = 0 for s _> 3/2.


6j q J ~

Figure 5.

Set r ----- r 6q,) /e) . We shall use the following notation: 61 = 6q, and pi(x) = d ( x , 6i). Finally, let

Clearly, {r is a partition of unity in a neighborhood of OU r. Now consider a point x E T.~ and its image 7, under the projection along radial arcs, in OU r. Suppose, in addition, that d (x ,~) <_ #/50. We need an estimate of the number N(x) o f j such that Cj(x) 5~ 0 and of d(5, qy) for such j . By Lemma 2.13, 6z and 6j are almost parallel. Therefore (see Figure 5) u and c = d(x, 6j) <_ 3e/2 are practicaUy equal. In the course of the proof of Lemma 2.7, we saw that v /u < 5 /# . Thus

v - Id(q~, 7 ) - r ( ~ ) l _ 8 ~ / # (2.16)

and

d('g, qj) g u + v < 2e + 8e/# <_ #/100 (2.17)

provided

e ~ #2/1000. (2.18)

It follows by a simple volume comparison, using the first inequality in (2.17), that for any x as above N(x) <_ c# -n, where c depends on n only. In particular, N(x ) is bounded independently of x and e by a constant that depends only on n and a.

We now make the following definitions:

H

= ~ d ( q i , 7 ) r i

=

= I = o ) . (2.19)

Tubes and Eigenvalues for Negatively Curved Manifolds 11

We claim that the set H has all the properties in the statement of the theorem. First we show that on every radial arc there is a point x such that r = 0 and d ( x , 5 ) < #/50. Start with a point �9 on the boundary of U-~. Let x_ , x+ be the points on the radial arc determined by 5, such that d(x+, 7) = d(5, 7) + # /50 and d(x_, 3') = d(5, 3') - #/50. We saw that, in the definition of r (y), for y in the segment connecting x+ with x_ , we only have to consider indices j for which d(~ ,q j ) <__ #/100. Now, r(x+) = r (~) + # /50 and v = [(r(~) - d(qj,7))[ < d(5, qj) < /~ /1 0 0 by (2.17). Therefore d(qi, 7) <- #/100 + r(~) and

9(x+) = r(~) + # / 5 0 - ~ d(qi, 7)r > #/lO0. i

Since {r is a partition of unity. Similarly, g(x_) < O. This proves that 9 has a zero on the segment connecting x+ with x_.

We now study the functions r = r We have

grad(r = r 1 grad(p/). (2.20) E

Therefore Ilgrad(r = O(1/e) . The gradient of Pi is tangent to geodesics emanating perpendicularly to 6i. By Lemma 2.13,

17~" grad(pi)[ < 3ca. (2.21)

The Hessian, Hess(q~i), as a quadratic form, is given by

1 r dpi | dpi + 1 r Hess(p/). (2.22) e2 c

On the other hand, on vectors perpendicular to grad(p/),

1 Hess(p,) = ]lgrad(pi)l I Hess(m )

is the second fundamental form of the distance tube ,_qp, of 6i. Since we have to consider only Pi E [e, 3e/2], we see from (2.15) that all eigenvalues r /of the Hessian of r satisfy

C 171 < - (2 .23 ) ~2 ~

where, as usual, c depends only on a and n.

12 ?. Buser, B. Colbois, and J. Dodziuk

We now estimate the gradient of g(x) for x E H.

grad(g(x)) = g r a d ( r ( x ) ) - Z d, g r a d / ~ ' ~ (x), , \E -jJ

where we write di = d(ql, 7). Choose i0 such that r ~ 0. Then

digrad ~ ( x ) = E ( d i - d i o ) g r a d ~ j C j (x). �9 i

The number of nonzero terms in the sum is bounded by a constant independent of x and e. Moreover, ~ j Cj > 1 and [di - dio[ < ce by (2.16). A simple calculation now shows that the sum above is bounded independently of c and is a linear combination of vectors grad(pj), i.e., is nearly perpendicular to grad(r). It is now that we choose and fix the value of e. In addition to satisfying (2.18), we choose it so small that

0 < c, < Ilgrad(g)[[ < c2,

with constants cl and c2 depending only on a and n, and so that the first assertion of the theorem is satisfied. Such a choice is possible in view of (2.21). We note that since e is fixed, the Hessian of Cj is bounded by (2.23).

The second fundamental form of H is given by

1 Ilgrad(o) II Hess(g).

In order to estimate the principal curvatures of H it suffices to bound Hess(o). We see from (2.15) that Hess(r) is bounded since we are working at large distances from 7. Thus, it remains to estimate Hess(C).

where, as above, we chose i0 such that r > 0. We saw above (2.23) that Hess(el) is bounded. Hess (~ j Cj) is also bounded since there is a uniform bound on the number of nonzero terms. Moreover, by (2.21), both the numerator and denominator of Oi/~, j Cj have bounded gradients. Using these properties and the fact that ~"~j Cj > 1, we see that Hess(r Cj) is bounded. It follows that the second fundamental form of H is bounded, so that H has bounded curvature. The bounds depend only on a and n. [ ]

Remark . The analog of this theorem holds for cusps. One has to use a Busemann function instead of r (x) and results of [23] about geometry of horospheres in addition to (2.15). The proof is very similar to the argument above and we omit it. [ ]


Coro l l a ry 2.24. For every point x E H, the injectivity radius with respect to the induced metric is bounded below by a constant depending only on the curvature bound a and the dimension.

Proofi This is an application of an argument of Klingenberg [28]. First note that for every x E H the radius of the largest ball around the origin in T~H on which the exponential mapping

is nonsingular can be bounded below by a constant depending only on the upper bound of the absolute Value of the sectional curvature. By Theorem 2.14(ii) this can be bounded in terms of a

and n. Therefore, a very short loop in H can be lifted to the tangent space at its initial point. A very short curve in H is short with respect to the metric of M . Thus such a curve is contractible

in M b y a "short" homotopy. Projecting along radial arcs, we obtain a "short" homotopy on H in view of Theorem 2.14(i). Therefore short loops lift to closed loops in the tangent space. Suppose now that the injectivity radius at x E H is very small. There exists then a short loop, consisting of two geodesic segments, which lifts to a closed loop in T , H . This is a contradiction, since the two geodesic segments lift to segments of different rays in the tangent space. [ ]

3. Eigenvalue estimates

The first theorem of this section is a generalization of results of [35, 14, 11] for manifolds

of dimension n > 4.

Theorem 3.1. Suppose M is a compact manifold of n >_ 4 dimensions with sectional

curvatures - 1 <: K < 0 . / f V = vo l (M) and ~i = )~I ( M ) is the first eigenvalue of the Laplacian A on M , then

c(n) Vd(~) '

where the constant c(n) depends only on the dimension n and d(n) = 6for n = 4, 5, d(n) = 3

for n = 6, 7, and d(n) = 2 for n >_ 8.

The method of proof of this theorem is patterned after [14]. The new ingredient is the estimate of the first Dirichlet eigenvalue of a tube. Under the curvature assumption - 1 ___ K < 0, the Margulis constant # = c(n), and the set M r = { x E M [ L(x) < # } is a union of finitely many tubes. Each tube is the union of maximal radial arcs as in Section 2. The estimate for the first eigenvalue of a tube referred to above is in terms of the length of the longest maximal radial arc. It can be carried out in all dimensions. It is here that the restriction n > 4 comes into play. We use Gromov's volume-diameter estimate [20] to obtain a lower bound in terms of the volume.

The theorem itself is false in two (cf. [13]) and three (cf. Example 4.4 below) dimensions.

For a vector v E T x M orthogonal to 7, at a point x of a short geodesic 7, T-r C M u,

denote by r(v) the length of the maximal radial arc in the direction of v. Define R = R ( 7 ) to be the length of the longest radial arc. Let A1 (T-r) be the smallest eigenvalue of the Laplacian

on T.~ for Dirichlet boundary conditions.

14

Lemma 3.2.

P. Buser, B. Colbois, and J. Dodziuk

/~ (TT) satisfies

> ( n - 2) 2

- 4R 2

Proof. This argument goes back to McKean [31]. Parameterize T = T 7 as follows. Every point p E T lies on a unique radial arc determined by a vector v in the unit normal bundle U N ( 7 ) at a distance t = d(p, 7). We take t and v as coordinates on T. Radial arcs are perpendicular to the distance tori { q E T ] d(q, q') = const } by Gauss' Lemma [18]. It follows that the volume element on the tube can be written as

dV = h(t, v)dt A dw, (3.3)

where dw is the volume element of U N ( 7 ) . By the standard volume comparison with a tube m the Euclidean space

1 Oh n - 2 ~ - > - - (3.4)

- t

We now estimate the Dirichlet integral of a function f E C~(T) vanishing on OT as follows. First note that Igrad(f)l ___ IOf/Otl. Therefore

Of 2hdt. fT lgrad(f)12 >-- fVN(.~) dw f~(~) Ot

Using the Cauchy-Schwartz inequality, integration by parts and (3.4) we obtain

n -- 2/ '~(~) >- 2---if- Jo f2h dr.

It follows that

> - h d t - - 2

fo "(~) Of 2 h dt >_ (n - 2) 2 fo ~(~) Ot 4R 2 f2h dt.

Integration over U N ( 7 ) yields

f r igrad(f)lEdv (n - 2) 2 f f2 dE. > 4.R 2 J r

[]


The following is an easy consequence of the Lemma and the volume-diameter estimate of [20].

C o r o l l a r y 3.5. If n >_ 4 then

c(n) I(T) _> vol(T)d(n) '

for every tube T C M u. Equivalently, for f E CI(T) vanishing on OT,

~ Igrad(f)12 _ c(n) vol(T) d(n) /7" f2,

where d(n) is the integer defined in the statement of Theorem 3.1.

P r o o f . According to [20], for every tube T we have

R _< c (n)vol(T) []

The next step in the proof of Theorem 3.1 is a C ~ approximation of the metric by a metric whose curvature tensor is C k.bounded (cf. [ 11 ]). Let g = ds 2 denote the metric of M . According to [1] or [3], given a positive integer k, there exists a metric ff on M with the following properties:

(i) �89 <_~ < 29

(ii) IIR-- llc c (n. k) (iii) g(x) _> c2(n, k) for x E M~,.

_g, Z denote respectively the Riemann curvature tensor and the injectivity radius for the metric ~.

Remarks.

(a) M u is defined in terms of the original metric.

(b) The property (iii) is a consequence of (i) and (ii) (cf. [I1]).

(c) The integer k = k(n) will be chosen and fixed depending only on the dimension of M as in [11].

(c) In view of the variational characterization of A I ( M ) , it suffices to prove Theorem 3.1 for the metric ~, modifying the constant if necessary. In addition, the ratios of L 2 norms with

respect to g and ff of a function or its gradient over an arbitrary open subset of M are bounded from above and below by constants depending only on n = dim M . The practical effect of this

is that we can replace g by a smoothed metric ff and assume that g has curvature tensor with a

16 1", Buser, B. Colbois, and J. Dodziuk

sufficient number of covariant derivatives bounded pointwise by constants depending only on the dimension. We shall do this and refer the reader to [11] for details. [ ]

Proof of Theorem 3.1.. Suppose that A = AI(M) < e / V d(~). We are going to obtain a contradiction provided e > 0 is sufficiently small. Let r be the normalized eigenfunction of the Laplacian, i.e., let r satisfy

m ~ 2 l-/~1(~ = 0

Mqbdg = 0

M ~2 = 1

]mlg rad(r = A.

The argument of [14] or [ 11 ] applies verbatim and shows that the oscillation of 95 on M \ M. is at most

Ol = C3(6 /vd(n) ) I /2V 1/2,

where c3 depends only on n. We now consider separately the following two cases:

(i) s u p z E M \ M ~ Ir k (ii) supzEM\M~, kb(x)[ < ot.

In the first case, r does not change its sign on M \ M~,. Hence a nodal domain D of ~ is contained in one of the tubes. Say D C T. It follows that ~ I D is a Dirichlet eigenfunction for D with eigenvalue A. By domain monotonicity and Corollary 3.5,

s c4(n) Va,~-'-S -> A' (D) ___ A~ (T) _> V~(------ S.

This is a contradiction provided e is chosen small enough.

To treat the second case, introduce the following notation. Let A = { z E M I r > a }, B = {x E M ] r < - a } , a n d C = {x E M I $(x)[ -< a }. Let r q~_ be equal to

+ a , ~ - a respectively. Clearly A and B are contained in My, ~b+ vanishes on 0B, and ~b_ vanishes on OA. By Corollary 3.5,

/A Igrad(r =/A Igrad(r > C4(Tt) r - - Vd(n) / A

- Vd(n)


Since ce = c3(6./vd(n))l/2V 1/2 and d(n) _> 2, one sees easily that

fACL > s and

fsr >- s r

The constants implicit in O(e) above depend only on the dimension. Adding these inequalities, we obtain

f c4(n) (l-/c Ir = JM Igrad(r 12 -> I

Since f c r = O(e) this is a contradiction provided e is sufficiently small. [ ]

The next theorem is a generalization of [6, (3.12)].

T h e o r e m 3.6. Suppose M" , n > 3, is a complete Riemannian manifold of finite volume V with pinched sectional curvatures-a 2 <_ K < - 1 , a > 1. The number of eigenvalues of the

Laplace-Beltrami operator of A of M in the interval (0, (~41): ] does not exceed c(n, a)V.

In [6] this was proved for n ----- 3, M compact and of constant curvature - 1 . We apply essentially the same argument using Theorem 2.14 to overcome difficulties caused by the fact that

tubes and cusps are not "round."

U~ N Theorem 3.6 will be proved by tiling the manifold M with sets { i}i=l so that N is at most proportional to the volume V, and so that for every i the first Neumarm eigenvalue #l (Ui)

of Ui satisfies

( n - 1) 2 ~ ( u ~ ) >

4

The lemma below will be used to construct these sets.

L e m m a 3.7. Suppose W is a compact Riemannian manifold with boundary, contained in the interior of a larger manifold W1 (if the boundary of W is empty, then W = W1 is compact). Assume further that the injectivity radius c(x) >_ eo for every x E W and that sectional curvatures satisfy IKI _< A 2 on W1. For c < ~0/2, choose a maximal set of points 7 ~ = {qj} of W satisfying d(qi, qj) >_ e for i 7s j . The Dirichlet domains

D~ = { x E W I d(x, q~) < d(x, qj) for i ~s j }


~ q j n *

qi

Figure 6.

are quasi-isometric to Euclidean e-balls. More precisely, if d(qi, OW) > 2e, then there exist a piecewise diffeomorphism �9 : Di ) B~ (0) C R" and two constants cl and c2, which depend only on eo and A, so that

c~(~*go <_ g <_ c2(~*go

on Di, where g and go denote respectively the Riemannian metrics of W and R n.

Proof. The proof of the corresponding fact in case of constant curvature metrics was given in [6, pp. 59-60]. We recall it briefly and point out how to handle additional complications. First note that it follows easily from the definition of 7 ) that the balls B,(q~) cover W, B,/2(q~) f3 B,/2(qj) = 0 for i # j . The Dirichlet domains Di are star-shaped with respect to their centers qi and, for every i,

B,/2(q,) C D~ C B,(qi). (3.8)

Let r~(x) = d(x, qi) and let ~ i be the gradient of r~. We claim that for a point x in a face of ODi, the angle ~ between 7"~i and the exterior normal n to ODi satisfies

71" ~b < / 3 < ~ , (3.9)

where/3 = / 3 ( A ) depends only on A. This can be seen as follows. Every face Fij of Di is given by the equation ri - r j = 0 for some j . Therefore

7~i - 7~j T / , ~ -

In - ni l

It follows that to prove our claim it suffices to give a lower bound for the angle (~ between 7-~i and 7"~j (see Figure 6). As we observed, d(q~, x) < e, d(qj, x) <_ e and e < d(q~, qj) <


2e. Let ~ be the triangle in the plane of constant curvature - A z whose edge lengths are d(qi, qj), ri, rj. If r is the angle in this triangle corresponding to r then clearly, r _< r r can be estimated using standard trigonometric formulae, thereby proving (3.9).

Now introduce normal geodesic coordinates y l y2, . . . , yn on B, (qi). Replace the metric g

on B, 0 (qi) by g0 = (dxl) 2 + " " q- (dx '~) 2. It is easy to see that the new metric is quasi-isometric to the old one and therefore (3.9) holds for 90 with a suitably adjusted constant. Now we use polar coordinates (p,/9) on B,o(qi ) to define a quasi-isometry of D~ with an Euclidean ball of radius c. For every/9 E S n-1 denote by r(O) the distance from the point of intersection of the ray determined by/9 with 0Di . Define ff by the formula

and use (3.9) (for the metric 90) together with (3.8) to verify that it is a quasi-isometry with desired properties. [ ]

We now use the lemma to construct the tiling of M. Let H denote the disconnected hypersurface consisting of smoothed boundaries of tubes and cusps constructed in Theorem 2.14. The complement of H in M consists of tubes, cusps, and one component "exterior" to all tubes and cusps. We call the closure of this component M1. Since H is a very small perturbation of OMthin, it follows from Theorem 2.14(iii) that the injectivity radius is bounded from below by a constant depending only on a and n at every point of M1. Let e > 0 be a small parameter whose value will be fixed later in such a way that it depends only on a and n. Choose a maximal set of points {Pi} C t t whose pairwise distances in H are greater than or equal to e. By Corollary 2.24 the injectivity radius of H is bounded below by a constant depending only on a and n. Thus, if e is sufficiently small then the Dirichlet domains Di (in H ) are quasi-isometric to Euclidean e-balls so that the constants controlling quasi-isometries depend only on a and n. We will simply say, in this situation, that the quasi-isometries are controlled. For every i, consider a hypersurface (in a neighborhood of Pl) through Pi perpendicular to radial arcs. Project Di onto this hypersurface along the radial arcs. Denote the image by Ei. By The- orem 2.14(i) this projection is a controlled quasi-isometry. Define the tile Ci, otherwise known as a piece of cheese, to be the union of all radial arcs emanating from Ei into the thin part of M. We shall refer to Ei as the base of Ci. In addition, define two tiles FiJ,j --- 1,2 as follows: F [ = { x E Mx [ x lies on a radial arc through Ei, (j - 1)e _< d(x, El) < j e }. It remains to construct tiles covering M1. Observe that any x E M1 belonging to the complement of F j has distance from H greater than or equal to e' = e s i n a , where ot is as in Theorem 2.14. Again, we choose a maximal set of points {qk} of M1 with pairwise distances greater than or equal to e' and apply Lemma 3.7. Denote the resulting Dirichlet domains by Gk. We summarize what we accomplished so far. The sets F j and Gk are quasi-isometric in a controlled way to Euclidean e-balls. All tiles are disjoint, except possibly for some Gk intersecting Fi j. Every tile contains an embedded ball of radius greater than or equal to e ' /2 and at most two tiles can overlap at a point in the interior of a tile. It follows, in particular, that the total number of tiles is of order Vie n.


The next step is to show that every tile has a large first Neumann eigenvalue #1. Since #1 of an e-ball in I~ n is c ( n ) / e 2, we can make sure that

( n - 1) 2 ( n - 1) 2 #l (P i ) >_ - - , # , (Gk) > (3.10)

8 8

To do so we have to choose c sufficiently small but the choice depends only on constants controlling quasi-isometries, i.e., only on a and n. Next we have to estimate #i (Ci). This is the crux of the matter. Fortunately, the estimate of Cheeger's constant given in [6, pp. 61-64] carries over almost verbatim. The only change required in the proof is to use the relative version of isoperimetric inequality [4] instead of the usual one. In any case, the argument of [6] proves that, if e is sufficiently small,

( n - 1) 2 #1(Ci) >_ (3.11)

4

The proof of this fact given in [6] is rather tricky and complicated. We believe that it is useful to give a simpler proof valid in the special case of a rank-one locally symmetric space.

L e m m a 3.12. Suppose C is a piece of cheese with the base E in a negatively curved rank-one locally symmetric space with the metric normalized so that the sectional curvatures are smaller than or equal to - 1 . The smallest positive eigenvalue tzl = #1 ( C) of the Laplacian on C with Neumann boundary conditions satisfies

/zl _> min(~,, (n -- 1)2/4),

where v is the first Neumann eigenvalue of the base E with respect to the induced metric.

Proof. Let r denote the distance of a point of C from E . We c~n parameterize C as I x E for some interval I (the interval is bounded if we are dealing with a tube and infinite in case of a cusp). In terms of this parameterization, the volume element d V can be written as f ( r ) d r A dA, where dA is the volume element of E and f ( p ) d A is the volume element of Ep = { x E C [ r ( x ) = p }. It is here that we use the assumption that the ambient space is rank-one locally symmetric. The Laplace operator in these coordinates takes the following form:

02u f ' Ou

5-7 + 7 N + A u, (3.13)

where fP is the derivative of f with respect to r and A. denotes the Laplacian with respect to the induced metric on E , . Let ~o be the eigenfunction of A on C belonging to #1 and satisfying Neumann boundary conditions. Define ~ by the formula

1 /E qo(r, .)dA. = vol(E-----5


Note that ~ ( r ) is simply the average of ~ on the cross section E , . Using (3.13) one can see easily that ~ satisfies A T + / z ~ = 0. We consider two cases: ~ -- 0 and 7 ~ 0. The second case is easier and we treat it first. Since ~ is orthogonal to constants, ~ satisfies fI ~(r)f(r) dr = O. It follows that ~(r0) = 0 for a point r0 in the interior of I . Thus ~ is an eigenfunction of A

on the domain C~ 0 = { x E C [ r(x) >_ ro } and ~ [ E~ o --= O. By a standard comparison - f ' / f >_ (n - 1) so that we can use a separation of variables argument as in Lemma 1 of [14] to conclude that/z > (n - 1)2/4.

that

If ~ -- 0, the function ~ is orthogonal to constants on every cross section E~. It follows

fE

where tJ, is the smallest positive eigenvalue for Neumann boundary conditions on E~. As above, we identify E . with E0 = E . Denote by g~ the metric of E . . Since the sectional curvatures are

negative, g. < go for r > 0. Hence Idul. > Idul0 for every function u on E , where 1-I~ is the norm of a cotangent vector computed in the metric g.. We can estimate now the Rayleigh-Ritz

quotient of a function u on E with respect to the metric g,:

fEldUl2, f (r)dA f~ldul~dA fE u2 f(r)dA fE u2dd --

if u is orthogonal to constants.

If d (~) denotes the exterior derivative along E , , we have by the inequality above

This proves the lemma. [ ]

We now spell out the choice of e. We choose it so that it is smaller than the injectivity radii of M1 and H , and so that (3.10) and (3.11) hold. We repeat that it is possible to make this choice depending only on a and n. After e is fixed, the number N of tiles can be estimated as

N <_ c(a, n)V.

Proof of Theorem 3.6. Let r r �9 - - , e g be the orthonormal sequence of eigenfunc- tions of A on M corresponding to 0 = )to < A1 <_ "-- < )~N respectively. There exists

f 6 span{ r r eN } for which fM f2 = 1 and fT f = 0 for every tile T. It follows

that fT Idfl 2 > # I ( T ) fT f2 for every tile T. Therefore

L 1 As>_ Idfl2 >-- ~ , Idfl2 + -{~'(Gk)k ~ f 2 + ~ ,,J


The factor 1 above comes from the fact that the tiles of type Gk and F/j may overlap. By (3.10) and (3.11),),iv > (n - 1)2/4, which proves the theorem. []

4. Examples

This section contains examples showing that certain assumptions in theorems of Section 3 are necessary.

Example 4.1. There exist a complete surface S of constant curvature - I and infinite

area with infinitely many eigenvalues below the bottom of the essential spectrum.

We construct S by gluing together Y-pieces, otherwise known as pairs of pants, according to a combinatorial scheme given by the Cayley graph of the free group ~'2 on two generators. A marked Y-piece Y is a compact surface with boundary 0Y = 71 t_J 72 U %, homeomorphic to a sphere with three disks removed, and equipped with a metric of constant curvature - 1 so that 7x, 72, 73 are geodesics. It is well known [7] that a Y-piece is determined up to isometry uniquely by the lengths li of boundary geodesics 7i. In addition, each boundary geodesic ")'i contains two distinguished points Pij, Pik which divide 9'i into two segments of equal lengths. Pij is, by definition, the point on 7i closest to 7j- When we talk below about gluing various Y-pieces, we mean attaching them along boundary geodesics of equal lengths so that the distinguished points on the boundary geodesic of one of the Y-pieces are identified with distinguished points on the other. There is still some ambiguity here; one can do this in two ways, but the isometry type of resulting surface is unique for surfaces we will be considering.

Let t > 0 be a parameter and let Yt be a Y-piece with ll = 12 = 13 = t. Consider the surface St built out of countably many copies of Yt by gluing them according to the Cayley graph/t" of /F 2. The spectrum of the difference Laplacian on this graph is bounded away from zero (cf. [12] and references therein). By a theorem of Kanai [27, Lemma 4.5] and Cheeger's inequality, the same is true for the Laplacian of the surface St. Because the group F2 acts on St by isometries, the bottom of the spectrum of St, A(St) is not an eigenvalue of finite multiplicity, i.e. belongs to the essential spectrum; ),(St) = #(St) , where # denotes the infimum of the essential spectrum. An easy test function argument (e.g. [13]) using the collar lemma [34] shows that l imt\o ),(St) = 0. It is also easy to construct for every pair s, ~ of positive numbers a piecewise C I quasi-isometry fist : Y~ ~ ~ which approaches an isometry as s ~ t. It follows that ),(St) is a continuous function of t. We can choose therefore a sequence 0 < ~1 < t2 < t 3 " " ~ to such that ),(St,) / ),(St0). Moreover, we can find for every i, a compactly supported function ffi on St, so that

f s ~b2----1 and ) , ( S t , ) < ~s Id~i l2<) , (St~ (4.2)

We now construct the surface S inductively as follows. Consider a "ball" in K of a sufficiently large radius so that the union of the corresponding Y-pieces in St1 contains the support of ~bl. Call the resulting surface TI. Attach to every end of T1 a Y-piece with boundary geodesics of lengths tl, t2, t2. Denote the surface so obtained by F1. Now take an "annulus" in K sufficiently


wide for the corresponding region in St2 to contain the support of (a translate of) r Attach this annular region of St2 to F1 and call the resulting surface T2. Construct F2 by attaching a Y-piece with boundary lengths t2,~3, t3 to every end. We continue this process and obtain an infinite sequence F1 C F2 C F 3 " - of surfaces whose union S has the desired properties. In fact, by the decomposition principle t15], # (S) = sup{ A(S \ F,) [ i = 1, 2, 3 . . . }, where A(S \ Fi) denotes the bottom of the spectrum of S \ Fi for Dirichlet boundary conditions. For large i, the metric of S \ Fi is quasi-isometric, with constants controlling the quasi-isometry approaching 1 as i tends to infinity, to the metric of Sto. Thus the difference between A(S \ Fi) and the bottom of the spectrum of the corresponding region of St o, tends to zero. Another application of the decomposition principle shows that #(S) = /z(St0) = A(St0). Moreover, by (4.2) and by the construction of S, we see that there exists an orthonormal sequence of functions {r with L 2 norms of gradients strictly smaller than .k(St0). This implies that the number of eigenvalues of the Laplace operator of Sto below the bottom of essential spectrum is infinite. [ ]

The next example is a variation on Randol's examples [33]. It shows that pinched curvature

has to be assumed in Theorem 3.6.

Example 4.3. For every n >_ 2 there exists a sequence of compact manifolds {Mk}k~__l such that dim Mk = n, Mk has constant sectional curvature - k 2/'~, vol(Mk) = v for a constant

v > 0 and the number ofeigenvalues of Mk in the interval (0, ~ 4 l? ) tends to infinity.

We begin as in the proof of Theorem 3.10 of [6]. Consider a compact hyperbolic manifold N with boundary consisting of two compact, isometric, totally geodesic submanifolds. Existence of N is obvious in two dimensions, simple examples in three dimensions are due to LSbell [29, 30] and Millson [32] proves that manifolds of this kind exist in all dimensions. Make a ring Nk out of k copies of N by gluing them together along boundary components (see [6, p. 52] for details). It is obvious that vol(Nk) = k . vol(N) and diam(Nk) ,,~ k / 2 and that for large k the "thickness" of the ring becomes negligible when compared to its length. Actually, the intuition for this example is that for large k, Nk looks more and more like a circle of length c �9 k and therefore has more and more small eigenvalues. We now rescale the metric gk on Nk. Mk is Nk equipped with the new metric

hk = k - ~ -gk.

Clearly, vol(Mk) = vol(N) for all k, and the curvature of Mk is equal to - k 2/'~ --~ - o c . Mk is even thinner than Nk and looks like a circle of length c. k 1-1/n ---, oc. This suggests that the number of eigenvalues of Mk in any interval (0, a] tends to infinity. The intuition is correct and can be justified by rescaling the inequality in [6, Theorem 3.10]. [ ]

The following shows that the restriction dim m > 3 is necessary in Theorem 3.1.

Example 4.4. For every positive integer m there exist a family { (Mk, gk) } k~=1 of compact Riemannian manifolds of three dimensions with uniformly bounded volumes and sectional curvatures satisfying - 1 <_ K < 0 such that )~m(Mk) ~ 0 as k approaches infinity.

24

Remark .

P. Buser, B. Colbois, and J. Dodziuk

(a) This cannot happen (cf. [35, 11]) if the curvature is bounded away from zero.

(b) By a variation of the construction below one can exhibit a sequence of complete noncompact manifolds with curvatures in [ - 1,0), with the essential spectra bounded below by 1, and with the number of eigenvalues in (0, 1) approaching infinity. []

Thurston's examples (cf. [37, 21]) of three-dimensional compact hyperbolic manifolds of bounded volume and arbitrarily large diameter are a starting point. In fact, there exists a sequence {Ark }k~=a of such manifolds converging to a finite volume hyperbolic manifold with one cusp. The cusp itself is the limit of tubes Tk in Nk surrounding a closed geodesic %~ of length 2Trek ~ 0. To simplify the notation we drop the subscript k of ek and write Te instead of Tk. We remark that for manifolds under consideration the tubes are round, i.e., all radial arcs of a tube have the same length and the radius Re satisfies

1 Re > c l n -

s

by Lemma 2.4. We modify the metric 9k of Ark on the tube Te by multiplying it by a suitably chosen function Ce. To define Ce we write the metric explicitly in terms of Fermi coordinates on T,:

he - dr 2 + cosh 2 r dt 2 + sinh 2 r dO 2,

where r is the distance from %, t is the arc length parameter along the geodesic, and 0 is the angle determining normal direction to %. Let f ( r ) be a C cr function on ]~ such that 0 < f ( r ) <_ 1 and

1 for r < 1/2 f ( r ) = 0 f o r r > 3 / 4 .

The conformal factor Ce is defined by the formula

C e = l + C e f ( ~ )

where Ce is a function of e such that

lim C, ~\oR--~ = 0

lira Ce = ~ . e \o

(4.5)

For e = ek, we define the metric 9~ = Ceh,. We abuse the notation slightly and write ge = 9k. Observe that 9k = hk on the complement of Tk. An easy calculation using properties of C, shows that the volumes of (T~, 9k) are bounded. Therefore (Nk, gk) have bounded volumes. On the set {r < we have 9k �89 = (1 + C~)hk. It follows that the curvature of gk on


this set is equal to --(1 -/- C,) -2, which tends to zero with e. Thus, for large k, (Nk,gk) contains a metric ball of large radius on which the curvature is very close to zero. By Cheng's theorem [10], Am(N~, gk) approaches zero. It remains to verify that the curvature of gk takes values in [--1, 0). The conformal factor is constant and greater than or equal to 1 on the set {r < U Thus, we _ 2R,} {r _> 43-R,}. have to consider only {�89 --< r _< ~Re}.3 Denote by K(a, g) the sectional curvature of a plane section cr with respect to a metric g. The formula for variation of curvature under conformal change of metric [19, (22), p. 97] shows that the difference between K(a, gk) and r hk) is given by a second degree polynomial without constant term in first and second derivatives (in our case first and second derivatives with respect to r) of ~ , = In r Using this and (4.5), we see that gk have bounded strictly negative curvatures. We rescale the metrics one more time by a constant factor to get curvatures in [ - 1,0). [ ]

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Received May 8, 1992

D~partement de Math~matiques, Ecole Polytechnique F6d~rale le Lausanne, CH-I015 Lausanne, Switzerland Forschungsinstitut ftir Mathematik, ETH-Zentrum,, CH-8092 Ziirich, Switzerland

Ph.D. Program in Mathematics, Graduate School and University Center (CUNY), 33 West 42nd Street, New York, NY I0036 USA

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Tubes and eigenvalues for negatively curved manifolds

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