Isoperlmet ric Ineclualit ies and the First Eigenvalue

of the Laplacian on a Riemannian Manifold Gordon Craig

Depart ment of Mat hemat ics and S tatist ics:

hIcGill University: Montreal

.\ thesis subrnitted to the Faculty of Graduate Studies

and Research in partial fulfilment of the requirements

of the degree of Master of Sciences.

July 1999

@ Gordon Craig, 1999.

National Library I*I of Canada Bibliothèque nationale du Canada

Acquisitions and Acquisitions et Bibliogaphic Services services bibliographiques

395 Wellington Street 395, me Wellington OttawaON K1AON4 Ottawa ON K1A ON4 Canada Canada

m u r ide Voire rafcirence

Our fi& Noire reterenca

The author has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant a la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or seIl reproduire, prêter, distribuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/fiIm, de

reproduction sur papier ou sur format électronique.

The author retains ownershp of the L'auteur conserve la propriété du copyright in t h s thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom it Ni la thèse ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation.

Abstract

In this survey, we will study the relationship betwcen isoperirnetric inequalit ies and

the first eigenvalue of the Laplacian on a Riemannian manifold. We will take t~vo

approaches. The Faber-Krahn inequality shows that a certain isoperimetric com-

parison theorem implies an eigenvalue cornparison theorem. Cheeger's inequality

gives a lower bound for the first eigenvalue in terms of an isoperimetric constant

the first eigenvalue, we will calculate h(M) for certain special cases and get usefui

approximations for more general situations.

Résumé

Dans ce mémoire, nous étudierons la relation entre les inégalités isopérirnétriques

et la première valeur propre du laplacien sur un variété riemannienne. Xous abor-

derons le problème de deux façons. L'inégalité de Faber-Krahn démontre qu'un

certain théorème de comparaison isopérimétrique implique un théorème de com-

paraison de valeurs propres. L'inégalité de Cheeger donne un minorant pour la . . 1 / . C ' c c ; ü ü CE c i : f i r u i I L - 1 . En pius

de tenter de comprendre la relation entre h ( M ) et la premitre valeur propre. nous

calculerons h ( M ) pour quelques variétés très simples et nous trouverons des ap-

prosimations qui nous seront utiles dans des situations plus généraux

Acknowledgement s

1 would like to thank my supervisor, Professor Toth, for his many suggestions, which

have greatly improved the clarity and readability of this thesis. I would also like to

thank my fellow students and my profersors here at McGill. as well as at L-QAhI

and Champlain Lennoxville. Particular thanks go out to Alain Bourget. for man?

stimulating conversations over my two years as a graduate student here. Gulhan

Aipargu, for teaciiing me iatex, Ai Forci. john Emerson anci Ciiaran Eucihiraja,

for starting me off in this direction at Champlain. and Professors Toth. Koosis.

Wolfson. Kamran, Hurtubise, Lalonde, Joyal and Pet ridis, for t heir guidance and

encouragement and for esposing me to a great deal of beautiful mathematics I

would also like ta thank the LIcGill math department, for not giving up on me too

soon. even though 1 certainly gave them reason to, and everone at Champlain. for

preparinp me so well and for giving me t ~ o of the best years of my life. I would like

to thank my parents for their love and support. and 1 s t but riot l e m . 1 want to

thank Mark, Xck. Laura, Greg. Juli. Dave, both Phils. Eric. Andrew. ShariE. Sick

and Astrid. mho made mu sis o a r s in Montreal such a great time.

Contents

1 Introduction

2 Isoperimetric Constants

3 The Faber-Krahn Inequality

4 Lower Bounds on h ( M )

5 Bounding X1(M) from Above in Terms of h ( U )

6 Calculat ing h ( J I )

7 Conclusion

Chapter 1

Introduction

This thesis will be an introduction to the interplay between isoperimetric inequal-

ities and the first nonzero eigenvalue of the Laplacian on a Riemannian manifold.

Ré will approach the subject from two different points of vietv: the Faber-Icrrihn

inequality and Cheeger's inequality.

Given a manifold hl and a positive nurnber c: an isoperimetric inequality gires

a lower bound on the ( n - 1)-dimensional measure(the orea) of the boundiiru of an-

open submanifold of volume v. Sote that an open submanifold must have the same

dimension as the ambient manifold. For esample: in Euclidean n-space. ne have

Theorem 1 (The Classical Loperimetr ic Inequality) For ang open subman-

zfold R of Rn equiped with the canonical metric:

zuhere Be is any bail such that l'(Bn) = I.'(R). Equality holds if and onfy if B,? is

a bail.

In general: isoperimetric inequalities are estremely difficult to est ablish. as t hey

make strong statements about both the local geornetry of the manifold (the volume

forrn) and the global geornetry (we are effectively taking an infirnum over a class

of submanifolds which could lie anywhere in the manifold.) The spaces for which

sharp isoperimetric inequalities are known are generally very special, such as simply

connected manifolds of constant sectional curvature (space forms) or surfaces. In

the first case, we have an extremely high degree of homogeneity, and in the second

me have a relatively simple structure.

The Faber-Krahn inequality relates an isoperimetric comparison theorem to an

analytic one. Vie have a manifold h l and a suitably chosen space form .\Ik of con-

stant sectional cumature k, dong with a fixed positive number u. Let $2 be an?

open domain in il1 such that V(R) = v. The isoperimetric comparison theorem

is as follows: an? such R must have that .4(aR) is greater than or equal than the

area of a geodesic bal1 of the same volume in hlk. The analytic one is that ,\,(fi).

the first Dirichlet eigenvalue of 0: is greater than or eqiial to the first Dirichlet

eigenvalue of a geodesic bal1 of the same volume in hln.. The idea behind the proof

is that if the isoperimetric comparison theorem holds, then ive are able to compare

JI to the more symrnetric manifold dlk in other w-s. This is an illustration of

the strength of isoperimetric inequalities. Note t hat the isoperime tric comparison

theorem mentioned above is still an isoperimetric inequality. The only special fea-

ture is that the lower bound for the area of aR is in terrns of the area of a geodesic

sphere in a different space.

Cheeger7s inequality will occupy most of our tirne. Although there are man'

lower bounds for X I in various special cases. Cheeger's is the only one whicli is

universal, i.e. which is applicable without any curvature bounds. ([Bu'!) Ir is

based on Cheeger's isoperimetric constant h ( M ) ? which ive will introduce dong

with another isoperirnetric constant and a Sobolev constant. We will then estnblish

that it is always possible to give both upper and lower bounds for the analytic

(Sobolev) constant in terms of the geometric (isoperimetric) constant. X e then

prove Cheeger's inequality, which is an analogous lower bound for X I in terms of

h(h1). We will also see that in general it is not possible to give a lower bound for

h(M) in terms of .A1. The reason for this is that h(J1) is not scale-invariant. To

prove al1 of these bounds, the key tool will be the CO-area formula.

In part 5, we ni11 further study the relationship between h(A1) and XI by seeing

under mhat circumçtances it is possible to give an upper bound for XI in terms

of h(h1). It turns out that mith certain curvature restrictions on iLI and the as-

sumption that it is compact, ive are able to do so. although the upper bound that

we obtain has a more complicated form that the lower bound we got in Cheeger's

ioequality.

CVe mil1 calculate h ( M ) for several special cases in part 6. but since this is more

or less equivalent to proving a sharp isoperimetric inequality for M. we only knom

how to do so for a relatively srnall class of manifolds. Since there are ~~ell-known

isoperimetric inequalities for space forms, me are able to calculate t heir Cheeger

constants with relatively little effort. The other manifolds nhich ive will esamine

are surfaces of constant curvature: the real projective plane and Bat tori.

Since it is extremely difficult t o calculate h ( M ) exactlu, in part 4 we will obrain

geometric lower bounds on h ( M ) for tno cases: iIf compact and of bounded cur-

vature, and 31 sirnply connected and of nonpositive curvature. The advantages of

these lower bounds are twofold; as ive11 as giving us easily calculated lower bounds

on XI ( A l ) . they also give us some insight into how the Cheeger constant is affected

by the geometry of M. Xnother useful consequence of these lower bounds is that

they show that h ( J i ) > O for large classes of manifolds.

In what follows. Jf will be a connected. n-dimensional Riemannian manifold.

possibly with boundary. If .Il has no botindary. it will be assumed to be complete.

and if i t has a boundary. Jl will be assumed to have compact closure. If 8.U = 0.

we adopt the convention that 111 n 8.11 = O. Since we will need both pointwise and

L* norrns of vector fields, to avoid confusion we ni11 denote the former by 1 1 and

the latter by II 11,. Pointwise inner products will be denoted by . and L2 ones by

( 7 ) -

Chapter 2

Isoperimetric Constants

Definition 2 Let d be the exterior derivative on A l , and let D be the cotmiant

derivative on ( M , g ) , eztended in the standard way t o one-fonns. Then, for any

f E Cm(l l f ) and p E M, Ddf defines a b i h e a r j o m on T p M . Ch cal1 Ddf the

Hessian of f , or Hess( f). W e define the L a p l a c i ~ n A to be the operator which

takes an3 f E Ca(dl) to

b is positive and formally self-adjoint a i t h respect to the L2 scalar producc

on Cx(M) . ([BGU]) If 8?11 + 0, we must specify boundary conditions for the

eigenvalue problem

These will be:

a Dirichlet boundary conditions: Oi O

Neumann boundary conditions: l a A r = O! where 8, is the normal deriva-

t ive.

Definition 3 W e define the spaces of admissible functions HLV ( M ) = { f E HL (x) f =

0) and HD(M) = { f E H:(I\.I)), where H:(M) is the completion of CF(31) with

respect to the HL Sobolev n o m on M.

Here, H,V(M) is the space of admissible functions for the Neumann problem

if aY # 0 and for the closed problem if 8M = 0, and H D ( h I ) is the space of

admissible func tions for the Dirichlet pro blem.

The reason for the condition JLl f = O in Hx(M) is that me want to eliminare

the case of io = c, which gives Xo = O. Ey restricting ourselves to f siich that

JhL1 f = O, we are taking the orthogonal complement of the constant functions.

s,(M) = inf ITf l d J I n (~.~[,jj*~q*-~

The Sobolev constant is the largest c such that the following inequalit:;

holds for al1 f E Ha. This special case of the Sobolev inequality shows that if a

function is in the L1 Sobolev space on JI. it must lie in L 5 (M). (.As long as

s,(JI) f 0' of course.)

Definition 5 CVe define the two isoperimetric constants

uhere S is an ( n - 1)-dimensional subm~nzfdd of ufhhich diuides into .\IL a n d - - - ~ i t h AT^ n M, = S.

For + @, we also define

where S = aM1 is an ( n - 1)-dimensional compact snbmanifold of M.

Remark: S o t e that in the definition of IN(M). Ive allow d M = O. %te also

that both the numerators and the denominators of the I'(AI) are of order n(n - 1)

in r , roughly speaking. This d l make them equivalent in a certain sense to the

s,(M), a relation that we will not have for Al (dl) and h ( M ) .

Theorem 6 (Co-area formula) Let f E C C ( M ) . Then Vg 'g EL (M) ,

d A t d t , L * g d v = L Lit1 ïv,

where T(t) = { p E Ml f ( p ) = t } and dAl is the induced measure on T ( t ) .

Furthemore, if V ( t ) = Y 01 ( { p E Ml f (p) > t } ) , then C'(t) E C" (Reg( f)) ,

where Reg( f ) is the set of regular values off, and

Proof: ([Cha]) The critical values o f f have measure O, by Snrd's Theorem. so T ( t )

is an immersed (n - 1)-dimensional çubmanifold of A l for almost al1 t E f ( S I ) . \lé

also have that Reg( f ) is an open subset of R.

Let (a, b) C R e g ( f ) , c E ( a , b) . Then let &, be the flow determined by the

vector field & restricted to / - ' ( ( a 7 b)).

@ ( p , t ) = # t - c (p ) defines a diffeornorphism of f -' ( c ) x (a: b) ont0 f -' (a. b). Since

d cl f (@(JI: t)) = Vf l o t& ) ~ - C ( P ) )

and f (9(pt c ) ) = cf ive have f (O(pt t ) ) = t for al1 ( p : t ) E f -L(~) x (a. 6).

Furthermore.

and

because it is parallel to the gradient of f and T ( t ) is a level surface of f . Becaure

a@, t ) is a diffeornorphism, we can take local coordinates (p, t ) on f - '(a. b). wliere

t E (a, b) and p E r ( t ) . B y the orthogonality relation above, we have

Note that we must have Iz12 in the lower right hand corner because $ is the

tangent in the t direction.

TT hus, locally,

= \ lde t (g i j ) dx, A . . . A ~ I , - ~ A d t

so n-e h a ~ e the CO-area formula.

Finally, note that

Q.E.D.

Proof: ([Cha]: [SE*]) ive mil1 focus on the Xeurnann case. Let S be $-en and

I*(.&) 5 V ( 1 4 ) . Then define

TI-here c, > O is chosen s - t . LL1 f, dl' = 0.

Since it is continuous and piecenise linear, f, E H,v(Jd). Now, as c -+ 0,

V(M1 ) where c = c,, which by dominated convergence is equal to -.

Furthermore, since I V d ( p , S)I = 1 , (because it is a distance function)

d i - .+- / dl. - ( p E M ~ , d ( p . S ) l ~ )

Once again using the fact thac I V d ( p , S) 1 = 1, ive can apply the CO-area formula

to get

and similarly for the other integral.

Thus, me get

because O 5 c 5 1 implies that

Thus, s s ( J l ) 5 2IX(dl) . Note that for sD ( d l ) . ne do not require thac Ji, f, d l* =

0: so Ive can take f, with d l t any n-dimensional submanifold of Jl. and c, = O. This

To prove the converse. let f E ccC(V) be given. and recall that cx(V) =

H'i (M) . Then there exists k such that for Ml = f-'(-cc. k ) . JI? = fdl(k.x). ive

have

i

Define fi@) = -min{f - ktO), 5 J p ) = mau{f - &.O) and I;(t) = I'({p E

dl 1 fi@) > t)). Then, by the CO-area formula,

Now, note that for any lunction g E Cm(=) with g 2 0,

= odwlq(tl tlGgl-' d(.4,)t d t . -

since d l being compact implies that I.,(x) = O. Kom, this is clearly eqiial to

(The last equality is by the co-area formula.)

In particular,

by a simple substitution.

i;owl define

n-1

We want to show that r,(x) 2 (*) di(^).

O, and

n- 1

But ?'(O) = ç e i ( ~ j = n- l

since C;(t) is decreasing. Ewluating the integral gives

so we have

Thus,

since 14v (.CI) 5 I i l . ( M i ) . By the triangle inequalitu, this is norv

Thus for al1 f E Ca(g), Ive have

and so since CCC (u) is dense in I-ln(M).

For sD (h l ) and ID(JI ) , we repeat the same proof. choosing Ml = f -'(O. cc). so

the proof is sirnpler.

Q.E.D.

We now introduce the Cheeger isoperirnetric constant.

Definition 8 We define, for M closed,

where S is a hgpersurface which divides into lLI1 and S. t. n = S.

If &LI f a, then we define

-4 (S) h D ( M ) = inf -

s L - ( M l )

As before, ive want to relate this geometric invariant to an analyric one. Recall

that the first nonzero eigenvalue(Dirich1et or Keumann) of a manifold of compact

closure is given by

x~,,(M) = inf {(Ar cf 1' d l * ) } a = D . X /fr Ha (ALI) ( f . ~ 1 f l2 d l ' - )

In the event that dl is not of compact closure. me can take equation ( 2.16) to be

the defintion of X1,,(.ll).

Then, w e have

Theorem 9 (Cheeger 's Inequality) ([Che])

Proof: ([SI']) Ré first prove the theorem for the Dirichlet case. Since really

is the first eigenvalue. the Courant nodal domain theorem ([Chai) tells us that the

associated eigenfunction does not change s i s . and so we c m assume that

$qD > O on M.

If there exists p > O such that

by Cauchy-Schwarz. Then we have

By the CO-area formula? if g E

h D ( M ) ( t l , ; ~ - lx f i :(L) d t )

If d M = B, ri-e note that by the Courant nodal dornain theorem. d1,.y = ol has

at most two nodal domains. Since J dl dl' = O. it must have esactly t w .

Take Ml to be the nodal domain of lesser volume. Then sarisfies the

Dirichlet problem on MI with eigenvalue XL,N(Al) = h l . and since 01 f O on .\IL.

XI must be the first Dirichlet eigenvalue of MI. Thus.

If 8M + @, to prove the inequality for the Neumann problem. n e proceed in the

same rnanner as we did to prove that I&\l) 5 s n ( M ) .

Q.E.D.

Remarks: It seems clear from al1 that we have done in this section that the

Keumann problem is the proper generalization of the closed problem to manifolds

with boundary. This is because whenever we look at the Neumann problem. me

take the compact manifold ;il into consideration, as opposed to the noncompact

manifold M .

I t is not possible to obtain the kind of equivalence between h,(.ll) and X ,,,(JI)

that we had for s , (M) and I , (M). We will corne back to this point in chapter 5 .

but the fundamental reason for this is that h,(.\.I) does not have the same kind of

"scale invariance" that I,(.CI) does. To illustrate this. note that A(S) is of order

rn - 1 , while \ * ( M L ) is of order rn. so if we take, for enample, the respective infimums

over spheres in 3-space, vie get:

but ( A ( S 2 ( r ) ) ) 3 ( d ~ r ' ) ~

inf = inf ., = inf 367 = 367. 00 (V(B3(r ) ) )? 0 ( 3 00

Uë close this section with a result of Eau's:

Theorem 10 ([Y]) In the definitions of haV(.\I) and Is(.\f), it suf ices t a let S

range over al1 hypersurjaces s. t. SIL and di2 are connected.

Proof: ([Y]) Lé ni11 only prove the theorem for Is(.U). since the proof for h .y ( J I )

is identical. We proceed by induction. Let II,(JI) be the isoperimetric constant

with the infirnum taken over al1 S such that dl l and Mi are connected. Then it

is clear that IN(M) 1 ï x ( M ) . We wish to show that VJII Ad2. irrespective of the

number of connected components, we have

Vie deal n i th this in a bit of a roundabout way. Because of the "min" in the

denominator, we work by induction on the number of connected components of S.

Let S = UZJi, where the Si are the connected components of S. Then. if

m = 1, S is connected and hence and hl2 must be too, so ( 2.19) holds.

Now, assume that (2.19) holds for m. Then we can assume that Ml is discon-

nected, because we know that (2.19) holds for MI and hl2 connected.

Let Ml = NU P, where i3:V = ~f ;= , Si, BP = ~ z < \ ~ SS,. Then. by hypothesis.

since M U(P U dl2) decomposes dl into two submanifolds which share the boundary

Therefore,

mtl

. s i > - (~k.(i\l))+[min(~*(i~)~\,~(~) + L - ( - u + ~ ) ) ] +

The second-to-1st inequality holds because % 5 1. and the l u t one holds

because al1 of the possible cases in the previous line are larger than both 1 -(.\IL)

and \-(JI2). Thus: we have (2.19). and n-e are done.

Q.E.D.

Chapter 3

The Faber-Krahn Inequality

Let Mk be the space Form

domein with piecewise Cff

Mk.

of constant curvature k, and let Q C_ iLf be an open

boundary. Let Bk(r) be a geodesic bal1 of radius r in

Theorem 11 (Faber-Krahn) ([Chu]) If, for all R AI, Cr(R) = I , ' (Bk(R))

implies that A(ôR) 2 il@& ( R ) ) , then 1,-(R) = I w ( B k ( R ) ) also implies that A! (!?) 2

A1 (Bk(R)), where A I zs the first Dirichlet ezgenualue.

Furthemore, if ue have that A(aR) = .-L(a&(R)) implies that Cl and B k ( R ) are

isornetric, then XI (R) = Xi(Bk(R)) aLo implies that C! and Bk(R) are isornrtnç.

Proofi ([Cha]) The proof uses û symmetrization argument. We will need several

functions, so let us try to simplify the notation as much as possible. Given f a Ca

function either on M or on Mk, let Vj(t) = V({pl f @) > t ) ) , r f ( t ) = f - ' ( t ) and

Al(t) = A(rl( t ) ) . Let $i be the first Dirichlet eigenfunction of R. Since cbl + O on

R, we cm assume that #i > 0.

Now, we Nish t o construct a radial function $ on Bk(R) such that I.;+(t) = l i, ( t ) .

We Vest show that V4, ( t ) is continuous. We oonly need to check a t the critical values

of 41, because it is COJ on Reg(#,).

Since V+, (t) is decreasing, al1 we have to do is to show that V(T( t ) ) = O for

dl t such that there exîsts p E r(t) with V411, = O, Le. that &(t ) has no jump

discontinuit ies.

If we take a chart of Riemannian normal coordinates centered at p, there rnust

be at l e s t one xi SUC^ that

To see this note that because #i is an eigenfunction, it follows that

Thüs, x c can takc a ncighborhccd !? cf p süch that if y" E !!: x d ni * Y + I - - 3,

so locally, by the implicit function theorem applied to $, p and q are coritained in

some (n - +manifold M. (With the same iV for al1 q E LI' with V& 1, = 0.)

Therefore,

Now, k t & ( r ) = I'(&(r)), & ( r ) = aBk(r) , = .A(Sk(r)) and T =

m a p E n $1 ( p ) . New, Vt E [O, Tl, define p( t ) to be given by

so R = p(0).

ÇVe want a symmetric function

to compare to +i , so we need to invert p(t). V4/a;, ( t ) is decreasing and continuous. so

since Vk(r) is continuous and increasing, p( t ) must be decreasing and continuous.

Furthemore, since V#, (t) and Vk(r) are a.e. CO", p( t ) must be too.

Therefore, p ( t ) has a continuous inverse

which is a.e. Cm and decreasing. Set

x d7-W)

where r(x) is the distance between z and the center of Bk(R) . --- c l ~ e have

Note tha t unlike V4',, ( t ) , I.i(r) is increasing, so a slight alteration of the CO-area

formula, along with the fact that the &(r) are level sets of the distance function.

gives

Since t = -1 0 p, w e have

Thus, n e get

(Since 7 is constant on the level surfaces of r)

= J; r ' (p(t) ) -4k(p(t) )p' ( t ) dt (by substitution)

- - T 2 r - O t v4,(t) dt (since y ( p ( t ) ) = t)

- - h $ : f l (by the CO-area formula)

Then we have for t E Reg($l)(i.e. almost al1 t E [O, Tl,)

1 JI"P,I d44t) ' ~ 4 k (dt)) (lb1 (tl \[m (By Cauchy-Schwarz)

Thus, we get

IV& l2 dV = iT e( t )d t (By the CO-area formula)

Putting al1 of this together gives

Now, assume that d(aQ! = A(ôBk! R)) implies that R and Bç(R) are isornetric.

If we have h l (n) = X1(Bk(R)), then we must have

forces Bk(R) and R to be isometric.

Q.E.D.

physical terms? the Faber-Krahn Inequality is telling us that if we are in a

space, i.e. one in which for a given volume, a ball rninimizes the boundary

areq then the shape which allows the smaliest standing waves is also a ball. Ué

see this by analogy with a vibrating membrane in R~ with the edges pinned d o m .

Let us normalize our units so that the wave velocity c is 1. The the wave

equation reads

But since Q is a standing wave, wery point of the membrane is performing simple

harmonic motion with angular frequency w , so V p E ER?

Thus, n e get

In other words, 2xfi is the lowest frequency mith which our membrane R can

vibrate.

Physically, for a membrane which is not circular, we expect some sort of inter-

ference around the corners to prevent nice, m a l 1 symmetric vibrations. which will

force the fundamental frequency of R to be large.

Using a slight variation of the Faber-Krahn inequality, we can prove

Theorem 12 (Obata-Lichnerowicz) ([Chu]) Let .ll he svch thnt Ric(J1) 2

k (n - 1) for sorne k > O . Then 2 nk, with equalitg iff -11 and :llk are isometn'c.

Proof: ([Cha]) Let dl be the first eigenfunction of 11 on II. and let and R2 be

its two nodal domains. with l '(fi!) 5 l ' (Cl2 ) . Then

\+(M) Let Id = -. This is less than or equal t o 1 by the Bishop-Gunther comparison

theorem. ([GHL]) Choose Bk to be a geodesic bal1 in the sphere J I k such that

Then by the Levu-Gromov inequality([GHL]).

with equality if and only if R1 9- Bk and AI & :\G are respectively isometric.

If, in the proof of the Fiber-Krahn inequnlity. \re define p ( t ) by \ & ( t ) =

,8Ck(p(t) ) , n-e will get Xl,D(SIL) 2 X L r D ( B P ~ with eqiiality if and only if R I & Bk are

isorne t ric.

Let f i be a hemisphere of A l k . Then by the domain monotonicity of Dirichlet

eigenvalues(if A C B, then /\1,D(.4) > XI,D(B).[Cha]) a e have

because on I&, the first non-constant spherical harmonic has eigenvalue nk. ([BGM])

If we have equality, thea QI and Hk are isornetric. This gives

and so we can repeat the whole process to show that Hk and R2 are also isometric.

Thus, !CI is isometric to hlk.

Q.E.D.

This theorern gives a sort of weak spectral rigidity result. It says t int for any

compact manifold of strictiy positive curvature Ricci curvature? if

n XI = - inf Ric

n - 1 A1

then 11.1 is a round sphere. Here, by infw Ric, we mean the smallest k such that the

hypothesis of the previous theorem is satisfied.

Chapter 4

Lower Bounds on h ( M )

In this section, unless noted, we will assume that &lI = 0. .-US) Recall tha t the Cheeger constant is given by h(M) = infs min(l..(,\.li ),\ .(,\ r 2 ) ) . where

h( w2 the infimum is taken over al1 Cm hypersurfaces S dividing !CI into MI and LI&. +- gives us a lower bound on XI(M), but we still do not know how to calculate it. I t is

obviously easy to get upper bounds on h ( M ) , but these are useless for the purpose

of estirnating XI (M). Another related problem is making sure that h(dl) > O. since

othenvise Cheeger's inequality is completely wit hout interest. ÇVe saw in chapter 2

that h(Rn) = 0, but this is because the ratio of area to volume is of order roughly

so taking larger and larger spheres allows us to make it as small as we please. r 7

On the other hand, for a compact manifold M. we can't choose to make a conves

h-ypersurface as large as m n t , so it would be reasonable to expect that h ( M ) > 0.

CVe must be careful, however, since these considerations are based on looking at flat

manifolds.

For instance, consider the following euample. ([BGàI]) Take iCI, to be txo unit

two-spheres connected by a cylinder of length L and radius r, suitably smoothed.

This d l give a "dumbbell" manifold. Consider a function f definpd on hlr which

assumes the value k on one sphere, -k on the other one, and is linear on the

cyhder , being constant on the cross-sections. Then

and

Thus,

O on the spheres V f = {

2- L on the cylinder

But then,

implies that h(Mr) + O as r -t O.

Another way of looking a t this is to say that the !CI, are tending tolvards a

disjoint union of two spheres, so Hodge theory requires the first two eigonvalues

to be zero. (This is hand-waving, since it could just as easily be argued that the

limit "manifold" is really two spheres held together by a string, and this space is

connected.) Note, however, that in the limit, the cuwature tends tonards -x

where the cylinder meets the spheres.

Under suitable restrictions on the curvature, however, we can obtain an upper

bound which depends only on the curvature bound, the dimension, the diameter

and the volume.

Theorem 13 ([Cr]) Let Ric(M) 1 (n - l ) k , k E R. Then if a(n) = I-'ol(Sn) and

d ( M ) zs the diameter of hl,

- s inh(J-k~) 2((r(n - 2) )d (M)

We adopt the convention that if k = O, G s i n h ( G r ) = r .

Proof: ([Cr])

If p E aR, define

Let f2 be a domain in :CI s.t an is a Cm hypersurface. Let p E fi.

Np to be the inmard pointing normal at p and let

where LI' is the unit tangent space at p.

Now, for p an arbitrary point in R, E Li,, define r ( ( ) = sup{s > 01 exp(t<) E

R,Vt 't [O, s]), c(<) to be the distance from

1 [€) = min(c(€). r (€1 ). Let 4 = (5 E LIpIl(E) = ~ ( 5 ) ) and w,

p t o its cut point dong exp(t<) and

- - where pp iç the canonical a ( n - 1)

rneasure on U'. w, is called the visibility angle of 82 at p because it measures the

proportion of geodesics leaving p along which we can "see" BR, i.e. the ones which

do not turn around somewhere inside Cl.

Let ;l be Liouville measure on UR, the unit tangent bundle, and do be the

induced measure on L W R = Upéan U:. Then we have:

Theorem 14 (Santalo's Formula) ([SI) Let f E L1(LTR) Then

Where Qt zs the geodesic flow along <: and 7i zs the projection of onto i t ~ . base

point.

Corollary 15 ([Cr])

CoroUary 16 ([Cr])

where W = infpEbfIi,.

ProoE Clearly, Wa(n - l)V(Q) 5 p ( U ~ ) . But

We must now calculate

But the inside integral is just the integral of (b, Np) over the hemisphere with mis

1 . (4 iv') dlir (&) = ki (cor 4 4) n(n - 2) J Cr;

Thus,

so we have

If we can just get

we Ml1 be done.

Q.E.D.

o w r bound for ij over al1 domains in AI with CO: boundary.

Lemma 17 ([Cr]) Let Ric(iCI) 2 (n - L)k. If S is a smooth compact hypersurfare

d z v i d h g Ad into hll und Al2, then

ProoE ([Cr]) Let O, = ( q E hl& = exp(ft), < E CG, t < c(<)), for some

p E Ml. If q E hl - 8,, take the Iength minimizing geodesic from p to q ? with

tangent vector Co, so q = esp(foto). We must have to 5 c ( 4 ) , since the ,geodesic is

length minimizing, and so 6;. Thus exp(cot) does not make it out of hll: and

q E Ml. Therefore ikf - 8, 2 il&, and 1L12 C 8,. \Ne then have

ÇVhere F(<, r) is the volume form for polar coordinates centered a t p. Note that

such polar coordinates are well-defined on 8,, because we are staying inside the cut

locus.

By the Bishop-Gunther theorem([GHL]), our curvature bound implies that

Interchanging 1\6 and !Cf2 gives us our result.

Q.E.D.

Putting it al1 together, we get

h ( M ) = inf =w) s min(V(M,), C'(M2))

( n - l)cu(n - l);i(M2) 2 inf

s L Y ( ~ - 2)d(iVI)

b'( JI) where MI is always s.t. \'(dl1) 5 \-(dlz). But then Lp(.1k) 2 7, so the previous

lernma gives

and we are done.

Q.E.D.

Corollary 18 If i ! is compact, h(itl) > O

ive can now get another lower bound for h ( M ) , this time for M simply con-

nected, but not necessarily compact. For the rest of this section, we will assume

that i\.l is simply connected with all sectional curvatures bounded above by -k,

k > O. Note that this forces M to be diffeomorphic t o Rn.

Theorem 19 (Hessian Cornpariaon Theorem) ([Y]) Let Mi, i = 1 1 , 2 , be

two complete, but no t necessari-! compact, manifolds, with */i : [O, a] i !Lri two

geodesics pararnetrized by arc length which do no t intersect the cut locus of their ini-

tial point. Let r i (x) = d(- f i (0 ) , x ) , x E Mi. I j for each fized t E [O, a] , X i E T-,,(,l~Lfi

with (Xi ,nfi ( t ) ) = O , llXill = 1, we have

where hli is the sectional curvature tenîor at -/,(t), then

For a proof, see Schoen and Yau. ([SY]) From this, we get

Theorem 20 (Laplacian Cornparison Theorern) Let Kr, be the space f o n

of curvatvre 4, k > O, let AI have sectional cumatuie bounded o b m e by -P and

rit!-, (y) be the distance function from some q E dl+ Then if r(x) is differentiable

at XQ E M and rLbf-, ( y o ) = r(xo), we have

Proof: Since the hypotheses of the preiious theorem are obviously satisfied. we

must have

where Xi and Y , are orthonormal frames for T',M and TyoiCLk respectivei.

Q.E.D.

domain with compact closure

= (n- ~ ) J ~ v ( R )

Theorem 21 ([Y]) If dl is simplp connected with al1 se ctional curvatzlres bounded

Recall that in chapter 2, v e saw that by taking a sphere in Rn, ive got that h(Rn) =

O, so this inequality is sharp.

We also have

Corollary 22 ([Y]) Let iCI be as in the previous theorem. If fi zs a simplg connected

domain of compact closure in M, then

Chapter 5

Bounding Xl(M) from Above in

Terms of h ( M )

In this section, we will assume that M is compact and that = a. We would like to show that there is the same sort of "equivalence" between

Al ( d l ) and h ( M ) that there is between I ( M ) and s ( M ) . In other mords, if Ive have

a continuously varying family of manifolds Mc, then h(Al,) 3 O iff XI(.U,) 3 0.

Since XI 2 7. as Xl(dl,) -+ O we must have h(MJ -+ 0, but we can construct

a counterexarnple to the converse([Cha]); take a sphere with two disks of radius e

punched out, and smoothly attach a cylinder to where the circles mere. Let L, be

the length of the cylinder.

If we take R i to be the cylinder, we get A(8Rl) = 4ae and kV(RI) - B L , e . so

zz &. This is not an upper bound on h ( M ) , because we must make sure that V ( W

V(Q1) 5 V(M - RL), or

But by taking É small, we can allon L, to be as large as n e wish. Thus. by

letting c -t 0, Ive can construct LCI, with h(Mc) O

We cm argue physically that the eigenvalues of E/I, converge to those of LU. The

idea is that heat d i h i o n on hlE tends to heat diffusion on M as c -t O.

FVe would have a difference between the diffusion processes if heat could diffuse

dong the "handle" as well as on the surface of the sphere. But as e + 0, the handle

becomes long and thin, so it will take a long time for the heat to diffuse along

it(because of its length) and not much heat will be able to travel aiong it at a-

time(because of its width.) Thus, most of the diffusion will occur on the sphere as

É + O, so the heat kernel of i\fe converges to that of M.

Since the heat kernel determines the spectrum, we have that

Note that on the handle, the curvature is not bounded below as E + 0.

Theoram 23 (Bt tser ) ( [Bul l ) Let Ric(iCI) 2 -k, k 2 O. Then

The roof of this is quite long. We will need several preliminary results first.

Lemma 24 (The Bochner Formula) ([GHL]) Let g E Cm(.\I). Then

1 - - l l ( l ~ r ~ 1 * ) = I f i e~s (~) l2 - (ilg12 + Ric(Vg. Vg) (5.3) 2

where IHess(g)I2 = &i((Hess(g))(,yj, Sk))2 for {Si) an orthonornaal basis of

nwP.

Since we are working pointwise, we can define H to be the linear map on TM,,

given by

Setting X = xy=t Xj, ive then have

Thus, from the Bochner formula,

and so if Ric(M) 2 -k,

Theorem 25 (Li- Yau) ([u Let Ric(h1) > -k, f E Cn'(i\l), f positire and

ProoE ([LI) Let g = log(e-lAf) for t E [O,T]. (Xote that f > O and ee tL is a

positive operator.)

Then g = = &(-rle-tAf) and, noting the divergence of a vector field by

v*,

Therefore,

From the Bochner formula, we get

Let H = tlVgI2. Shen Ive have

H Since H = t $ l ~ ~ ( ~ i IVg/? means that t $ l ~ g l ' = H - T.

Now, let 1 = tg , so by equation ( 5 . 0 , differentiated. we get

so we have

Consider G = H - al = t(lVg12 - ag). PVe have by equations ( 5.9) and ( 5.10)

Now, G is defined on AI x [O, Tl, a compact set, so it must reach its mwimum

at some point (XO, ta) E il1 x [O, Tl. For now, assume that G(xo , to) > O. Then,

since G = t(lVg12 - ag) > O , tO > O and so at (xo, to) we must have VGI(,,,, = 0:

d(x0, ta ) = O and 4Gl~l,,t,! 2 O. (A11 this can be seen by taking a normal coordinate

chart at jxo, toj and appiying the corresponàing resui~s for RE x Rj.

Thus at (xo? to), we have, by equation ( 5.12)

Since Ag = - j + lVs12 and 9 = -5 -+ F, we get at (zO, ta)

Setting J = q, we grt

Thus, for generai (x, t) E W x [O, TI, Rie have

na2 (Since cr > 1 and t J ( x , t ) 2 O)

(1 + (a - 1) tJ )

If rnau,bfX [O,=] G ( x , t) 5 O, this inequality holds trivially.

Since

we get

Q.E.D.

Before going any further, Ive need a lemma about the heat operator.

Lemma 26 Let f E COC(M). Then we have

If f > O, we also haue

l l e - t A ~ l l f x 5 Il! Ilcc

Proof: For the first equality, conçider

Thus, J,, e-tA f dl/' is a constant with respect to t , and so it is alvays equal to its

initial value J,5f f d V .

For second inequality, note that

since h and edt" are positive operators ([RI) and f > O. Thus, e-'" f is decreasing

with time. Again using the facts that f > O and that ëCA is a positive operator.

we see that e-'" f 2 O and so we are done.

Q.E.D.

We noom use the Li-Yau inequality to obtain:

Lemma 27 (Varopoulos) ([LI) Let T > 0. f E C'"(AI) and f > O on -11 Then

i i n < t < ~ , - -

Proof: ([LI) Let o = 2 in the L i -hu inequality. so Ive haïe

(Note that e-tA is a positive operator.)

Thus,

Let f)- = m a ~ ( - A e - ' ~ f. 0) and ( l e - L A f )+ = mas(1e-'" f . 0) . Since

l e - t " f d i - = O. we must have

L [ ( & - t ~ f ) - d l - = 1 ( l ë t L f )+ d l - .\ I

By equation ( 5.17) ?

Therefore,

by equation (5.14). Since f is positive, we have

Since f E Ca(iLf),

The last step is because h and e-tA are self-adjoint and cornmute. This is in

turn

Returning to L i - b u , we have

Taking Lm norms gives

by inequalities (5.13) and (5.23).

Thus, we have

Q.E.D.

Now we can prove Buser's result.

Proof of Theorem 23: ([L])By assurnption. k > O. Shen in Varopoulos's

inequality, we let T = f . This rnakes C = (6n(l + k ~ ) ) ; = = C ( n ) . Let

g E COE(M), llgllm 5 1. Then we have

Thus, we have

I l f - e - Y Ili =

i

Now, let R be a domain with Cm boundary in M. If we take smooth approsima-

tions to its characteristic function Xn, we will get Ji\[ [Vxn[ dV = A(3R) (We did

something similar in the proof of the equivalence of I ( M ) and s(hf" in part 2.) '

We now have by (5.27)

because O 5 ë-tAXn 5 1.

Since

Now, note that for al11 E CCE(M) such that J f dV = O , we have

Where the $i are the orthonormal eigenfunctions of A. We start the sum at i = 1

instead of i = O because (f, 1) = 0.

Thus, we get

We now have

11 -a ( "(n) ) -tA ( "(" ) 11' ~ l e y ~ n l l a = e? xn- - I. ( M )

+ e T - V(") *

The third term is

and the second one is

We can apply (5.28) to the first term, since

CVe then get LA ;~i. 11 ~ ( 0 ) 11' v(ni2

Ile-Txnll? <- e Xn - - V ( M ) , + F(zj

we have

V (M)

SO

for t E [O, i ] (5.30)

Given an, to pet Cheeger's constant, we always take the part 114th the smaller v W) volume, so V ( Q ) +. Taking the inf me get

If XI 2 k, setting t = &

On the other hand, if XI 5 k, take t = to get

To get the last line, we used the power series expansion of ex. This gives

Either way, Ive are done, taking

Q.E.D.

Remark: This bound is actually sharp in the sense that we can find families

of manifolds lLl, s.t. as h(Mt) -+ cm, XI (Mt) is 0 ( h 2 ( M t ) ) and families of Riemann

surfaces for which Xl(Mc) 2 Constant -h (MC) as h(Mc) + O. The second case is

quite complicated, but the first one can be illustrated by taking square Bat tori.

of generator length t , which give X1(iWt) = 5. h(illt) is roughly of order i: so as

t -t O (i.e. as X 1 (Adt) + cc),

Chapter 6

Calculat ing h ( M )

In this chapter, we Ivill calculate h ( M ) for several manifolds. As we noted previ-

ously, this is a very difficult problem, since it is more or less equivalent to obtaining

a sharp isoperimetric inequality for M . Not surprisingly, the manifolds for which it

is possible to determine the Cheeger constant exactly are very simple: space forms

and surfaces. We begin with the follonring result, which ive will not prove.

Theorem 28 ([q) In the space forms, arnong all hypersurfaces enclosing the sume

volume, a geodesic bal1 has the least area.

This is very reasonable, for three reasons which are exceptional to space formç.

First of d l , they are completely homogeneous, so the location or position of a given

hypersurface will not affect its area or the volume contained Mthin it. This tums

the isoperimetric problem into more of a local problem. Secondly, space forms are

simply connected, so there is a restricted nurnber of possible candidates for the

minimizing hypersurface. (See the case of the 0at torus below for an example of

how homotopically non-trivial hypersurfaces corne into play.) The third reason is

that the volume form is very simple, and so calculations are easier. Note that

weveaker versions of the first and third conditions apply to symmetric spaces. but the

isoperimetric problem is still unsolved for general symmetric spaces.

R o m the theorem, me get

Corollary 29 Let a(n) = V ( S ) . Then

Remark: Here, al1 three manifolds have their canonicd metrics.

in these three spaces. Now, note that the hinctions

are decreasing in r for each fked k. (This can be seen by tâking their derivatives.)

Thus, to calculate h ( M ) for each of the three spaces, Ive simply have to set r equal

to half the diameter of M . For Sn, this gives the area of an (n - 1)-sphere over the

volume of a hemisphere. For the two other cases, ive get:

nrn-L h(Rn) = lim - = O

r-bm T"

and (sinh r)n-l

h(Hn) = lim r+OO Ji (sinh s)"-' ds

4pplying 17Hôpitai7s rule to this gives

(n - 1) sinhn-* r cosh r = Iim

~ + O Q sinhn-l T

(,n - 1) cosh T = lirn

r+OO sinhr

Q.E.D.

For the rest of this chapter, we will deal exclusively nith compact surfaces

without bound- We wiil denote one-dimensional measure by L and for two-

dimensional measure we d l use an A. By a domain, we will mean an open, con-

nected set with COD boundary. What follows is based on the work of Hass, Howards,

Hutchings and Morgan on the isoperimetric problem on surfaces. ([HM], [HHM])

ive begin by proving the existence of and characterizing the domains R such that

Definition 30 Let R be an open set in Ad with smouth boundary. R is said to be

l o c d y convex if fo r euery p E an, there is an open bal1 U arovnd p such that U n R

is geodesicallg conuez.

Theorem 31 Let hl be a compact surface. Then there ezists a locally conuex do-

main il with srnooth bounday such that

ProoE Let 3 be the set of al1 smooth curves(possib1y consisting of several

connected cornponents) which divide il1 into two domains. Since Af is compact, if

c E 3, then c is the disjoint union ut, c,, where each ci is diffeomorphic to S' and

k is less than or equal to the genus of M.

For each q, define Ri to be the domain bounded by q. Then there is a seqiience

of ci E 3 such that

lim U S ) = h(!\f) a-+= min(A(ni), A ( M - ni))

Taking a subsequence if necessary, Arzela-Ascoli tells us that c = .limi,, ci is

smooth. Since 3 consists only of a finite number of homotopy classes, by taking

another subsequence, we can assume that al1 of the ci are homotopic, and hence

that c is homotopic to ci. In particular, even if M is non-orientable, c bounds a

region.

Let 0 be the interior of c. c cannot intersecc itself, R or Ad - Q on a set of

nonzero length, because otherwise we could delete that segment and end up with

a shorter curve bounding a region of the same volume, a contradiction. Thus the

only possible self-intersections of c are of the forrn

and there can only be a finite number of them, since c is smooth and compact.

convex. LVe have two possible cases:

In the first case, let p be a point of dR at which R is not locally convex. If p

is not an intersection point of c with itself, then by taking a srnail ball U centered

at p and then taking the convex hull of U n R ne will increase the area and strictly

decrease the length.

If p is an intersection point, take another small ball centered at p, and convesify

as in the figure below to again increase the area and strictly decrease the length.

45

In either case,

tively to fi and E,

take the bal1 U srna11 enough that after altering R and c respec-

we still have ~ ( 0 ) 5 9. We then have

a contradiction.

If .4(R) = y = A(M - R), and neither R nor d l - R is locally conveq then

repeating the çame process as above at one point where R is is not locally conves

and at another where M - 0 is not locdly convex will give us a curve C dividing

i1f into fi and M - fi. We d l have L(E) < L(c) , and we cm choose the sizes of

the two balls such that A@) is close enough to A(R) that we have

another contradiction.

Q.E.D.

Corollary 32 If T is a rectangularpat toms vzth generator lengthv a and b? a 5 b:

then h(T) = $

ProoE The possible bcundary for our minimizing regioii R is either one closed.

simple curve or two non-intersecting homotopical1 y non-trivial curves. Furt hermore.

the region which they bound must be locally convex. In particular, this means that

for the first case, we must take the interior of the curve, defined as the part that

lifts to a disjoint union of bounded regions in the universal cover.

We begin by getting a lower bound on 9 for an a simple closed curw. In

this case, 0 is isometric to a bounded region in the plane, and so by the classical

isoperimetric inequality, dong with the fact that V ( T ) = ab,

L(an) TT .i(n) 3 inf -

{ r l m ~ ~ ~ } ?Tr2

Note that this is just a lower bound, since a circle of radius might be too

wide to fit into T withou t self-intersections.

Taking the infimum over al1 regions bounded by two curves is easier, since we

can find two curves of shortest length in T bounding a region of area 9 sirnply

by chosing two lines parallel to the shorter generator and a distance $ apart. This

gives

Since a 5 b and 4 < 2&, we are done.

Q.E.D.

Now might be a good time to see why the methods that we are using do not

extend to higher dimensions. Essentially, what we used to prove theorem 39 is the

fact that local convexification decreases area and increaseç volume, and that it is

an effective way to eliminate self-intersections. To obtain a counterexample to the

first statement, take a plane with a fold in it, as in the figure below, and consider

the domain contained below the plane.

If we take a point a t the bottom of the trough, and convexify a neighborhood

of it, we will have a bubble around the point. This increases volume, but it also

increases the area, no matter hon small the neighborhood is. As for the second

statement , in higher dimensions, self-intersections will not occur at points, but

rat her dong (n- 2)-dimensional hypersurfaces. Thus, we would have to take tubular

neighborhoods of the self-intersections instead of just balls around them.

The two steps that we used for the 0at toms once we had proven existence of a

minimizing region are also much more complicated for higher dimensions. First of

all, we used the fact that the only possible boundaries are either one or two circles.

If the genus of M were higher, an could contain more circles, but a t least in two

dimensions we only have one possible topological type for each component of the

boundxy. In three dimensions, we have an infinite number of possible boundaries,

corresponding to the closed, compact surfaces, and in higher dimensions we do not

even have an exhaustive list of the possibilities.

The second fact that we used was that the hypersurfaces which minimize length

in a surface are geodesics. A g e a t deal is known about geodesics, but relatively

little about minimal surfaces, their higher-dimensional counterparts. Thus. even if

we n-ere able to figure out what topological type a boundary had, we might still be

unable to minimize its area. ([BI])

We close out this section by calculating the Cheeger constant of the projective

plane (wit h the standard metric.)

Theorem 33 h ( ~ P 2 ) = fi

Proof: -4s before, we know that the minimizing region Sl must be locally conves.

Since it bounds a domain, a R must be homotopicdly trivial. Thus. lifting ro the r-tRP2, - sphere will give us a locally convex domain i! with volume less than ., -

v(S21 - = T. Btit we know that arnong domains of fixed volume in the sphere. it is 4

the geodesic bal1 which minimizes surface area. Therefore

2;r sin r h(RP2) = inf

{r\?r(l-cw r ) i r } 2 ~ ( 1 - cos T )

Since 2i(;?& r) is a decreasing function of r , we simply have to take the largest

r for which 27r(1- cos r ) 5 r. This turns out to be r = 9 , so

Q.E.D.

Remark: Note that h(s2) = 1 < 6 = h(R.P2). This is consistent n i rh the

fact that every eigenvalue of R P ~ is also an eigenvalue of s*. Remark: Although at tirst sight this argument would seem to work in higher

dimensions, we used Theorem 31 to obtain existence of a minimizing domain R.

Chapter 7

Conclusion

Our main goal in this thesis was to illustrate the interplay betmeen isoperimetric

inequalities and the first eigenvalue of the Laplacian on a manifold. This nas done

by proving Clieeger's ineqcality and the Faber-Krahn inequality, which gave us

good geometric lower bounds for XI (111) in terms of isoperirnetric constants. The

main value of this is that these lower bounds, in particular Cheeger's, are valid in

an extremely wide range of circumstances. In chapter 5, we established an upper

bound for Al (M ) in terrns of h ( M ) under certain conditions. This showed us that

it is also possible to majorize the first eigenvalue in term of isoperimetric constants.

The other two sections were concerned with getting a grip on the Cheeger con-

stant, which is obviously quite difficult to cornpute. Although we were onlp able

to calculate it for a very small class of manifolds, in chapter 4 we got some decent

lower bounds.

Bibliography

[BGM] Marcel Berger, Paul Gauduchon and Edmond hfazet. Le Spectre d'une

Variété Riemannienne. (Lecture Notes in Mathematics? 194). Springer-

Verlag, Berlin-New York, 1974.

[Be] Pierre H. Bérard. Spectral Geometry: Direct and Inverse Problems (Lec-

ture Notes in Mathematics, 1200, Springer-Verlag, Berlin-'iew York,

1986.

[Bull PeterBuser."OnChee,~er'sInequalityX~~~." GeometryoftheLaplace

Operator. (Proc. Sympos. Pure Math., ,XXXVi), pp. 29-77, American

Mathematical Society, Providence (R.I.) , 1980.

[Bu21 Peter Buser. "A note on the isoperimetric constant!' Ann. Sci. Ec. Sorm.

SUP. (4) 15 (19821, pp. 213-230.

[Cha] Isaac Chavel. Eigenvalues in Riemannian Geomety. Academic Press. Or-

lando, 1984.

[Che] Jeff Cheeger. "A lower bound for the smallest eigenvdue of the Laplacian."

Problems in Analysis (Papers Dedicated to Salomon Bochner, 1969). pp.

193-199. Princeton University Press, Princeton (N. J .) : 1970.

[Cr] Christopher Croke. "Some isoperirnetric inequalities and e igendue esti-

mates." Am. Sci. Ec. N o m . Sup. (4) 13 (19SO), pp. 419-135.

[GHL] Sylvestre Gallot, Dominique Hulin and Jacques Lafontaine. Riemannian

Geometry. (Second Edition), Springer-Verlag, Berlin, 1990.

[HM] Joel Hass and Frank Morgan. "Geodesics and soap bubbles in surfaces."

Math 2. 223 (1956), pp. 185-196.

[HHM] Hugh Howards, Michael Hutchings and Frank Morgan. "The isoperimetric

problem on surfaces." Amer. Math. Monthly 106 (1999), pp. 430-439.

[LI Michel Ledoux. "A simple analytic proof of an inequality by P. Buser."

Proc. AMS. 121 (1994), pp. 951-959.

[Mi] John Milnor. Morse Theo y. Princeton University Press, Princeton (3. J.).

1963.

[Mo] Frank Morgan. Geometric Measure Theoq: A Beginner's Guide. (Second

Edition.) Academic Press, San Diego, 1993.

[O] Robert Osserman "The isoperimetric inequality." Bull. AhIS 84 (1978).

pp. 1152-1235.

[RI Steven Rosenberg. The Laplacian on a Riemannian ~Uani fo ld . Cambridge

University Press, Cambridge (U.K.), 1997.

[SI L. A. Santa10 Integral Geomet y and Geometric Probubility. r\ddison-

Wesley, Reading (Sf as.) , 1976.

[SY] Rick Schoen and S. T. l'au. Lectures on Dzfferential Geometri~. (Confer-

ence Proceedings and Lecture Notes in Mathematics, V. 1)? International

Press, Boston, 1994.

VI S. T. Yau. "Isoperirnetric constants and the first eigenvalue of a compact

Riemannian manifold." .4nn. Sci. Ec. Norm. Sup. (4 8 (1975) ? pp. 487-507-