Graphs with parallel mean curvature and a variational problem in conformal geometry

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T H E B R IT IS H L I B R A R Y

D O C U M E N T SUPPLY CENTRE

TITLE

CURVATURE AND A VARIATION AL

GRAPHS WITH PARALLEL M EAN

PROBLEM IN CONFORMAL G E O M ETRY

A U TH O R

Isabel Maria da Costa Salavessa

IN STITU TIO N

and DATE

University of Warwick

A ttention is drawn t o the fact that t h e copyright o f

this thesis rests with its author.

This copy o f the thesis has been supplied on condition

that anyone w h o consults it is u n derstood t o recognise

that its copyright rests with its author and that n o

information derived from it may be published w ithout

the author’s prior written consent.

T H E B R I T IS H L I B R A R Y D O C U M E N T SUPPLY CENTRE Boston Spa. Wethertoy West Yorkshire

United Kingdom R E D U C T IO N X __________

20

GRAPHS WITH PARALLEL MEAN

CURVATURE AND A VARIATIONAL

PROBLEM IN CONFORMAL GEOMETRY

by

Isa bel M aria da C osta Salavessa

December 1987

A b stra ct

This thesis essentially deals with tw o basic problem s, one in Rieinannian, the other in C onform al Geom etry, described in P art I resp. Part III. P art 11 can be considered as an interlude serving as a sort o f bridge between Riemannian and Com form al G eom etry.

The main result o f the first part, form ulated in Corollaries 1.1.1 and 1.1.2 o f Theorem 1.1, states that any graph 1 / C M x N o f a map / : M —* N between Riemannian m anifolds, with parallel mean curvature, is minimal, provided M is com p act or n on -com pa ct with zero Cheeger constant. This result generalises the case M — IRm, N = JR, independently treated by E . Heinz, S. S. Chern, and H. F . Flanders. M oreover, Theorem 1.2 and P rop osition 2.3 show that, for M the m -hyperbolic space — thus with non-zero Cheeger constant — there exists a real- valued function / , the graph o f which is a subm anifold o f M x ] R w ith parallel mean curvature H satisfying ||1?|| = e, where e can be any positive constant less than or equal to the ratio o f the Cheeger constant and the dimension m . Furthermore, the behaviour o f the mean curvature o f a graph is studied in som e special cases.

The second pa rt deals with the problem o f finding a criterion for an im m er­ sion between Riemannian manifolds to b e a conform al one. Sufficient conditions on the mean curvature, tension field, and ratio o f given and induced volum e el­ ements in the im m ersed manifold are derived in T heorem 1. T h ereto, a special, “alm ost conform al” vector field is in troduced, which also allows the obtainm ent o f a Liouville-type theorem for harmonic maps.

Forew ord

It u a pleasure to thank Professor Jim Eells for his constant encouragem ent and his liberal attitude towards research. M oreover, he offered me several very valu­ able suggestions o f problem s to study. I am also in debted to D r. M a rco Rigoli for introducing me to the somewhat m isterious, old-new field o f conform al geom ­ etry. Further, I should mention the helpful conversations with D rs. D . M . Due and R . Tribuzy. T h e Calouste Gulbenkian Foundation in Lisbon is thanked for its financial support during m y stay abroad and the Faculty o f Sciences o f the Uni­ versity o f Lisbon for perm itting m y leave o f absence. Finally, I dearly thank my husband George R u p p , w h o lost tw o m onths o f his w ork in form atting m y thesis with and supported me decisively in its realisation.

Contents

A b s t r a c t ii

F o r e w o r d iv

C o n te n t* v

I G R A PH S W IT H PAR ALLEL M E A N CU RVATURE 1

0 G E N E R A L R E M A R K S A N D N O T A T I O N S 2

1 T H E M E A N C U R V A T U R E O F A G R A P H 10 1.1 In tro d u ctio n ... 10 1.2 The General C a s e ... 12 1.3 Co-Dim ension O n e ... 21 1.4 Graphs o f Isom etric Immersions, Conform al M aps, Riemannian Sub­

mersions, and H arm onic M o r p h i s m s ... 24

3 S T A B I L I T Y O F A M I N I M A L G R A P H A N D A G E N E R A L I S E D E Q U A T I O N F O R N O N - P A R A M E T R I C H Y P E R S U R F A C E S W I T H C O N S T A N T M E A N C U R V A T U R E 36 2.1 Some Remarks on the Stability o f a M inim al G r a p h ... 36 2.2 The Equation fo r a Non-Param etric Hypersurface o f (A f x

JR,

g x h)

with Constant M ean Curvature: Som e Remarks on Regularity o f S o l u t i o n s ... 40 2.3 Existence o f G raphs o f Functions on the m -H yperbolic Space with

Given Constant M ea n C u r v a t u r e ... 51

II

C O N FO R M A L A N D ISO M E T R IC IM M ERSIONS

O F R IE M A N N IA N M AN IFO LD S [Ri-Sa/87]

62

1 In tro d u ctio n ... 63

2 Preliminaries and F o r m u l a e ... 63

3 M ain R e s u l t s ... 68

R EFER EN CES 73

III A V A R IA T IO N A L P R O B L E M A N D A RELATED

B E R N S T E IN -T Y P E T H E O R E M IN CO N F O R M A L GE­

O M E T R Y

75

0 IN T R O D U C T IO N 76 1 T H E C O N F O R M A L G E O M E T R Y OF SU B M A N IF O L D S OF 5 " 78 1.1 The G eom etry o f the M öbiu s S p a c e ... 78

1.1. A T h e Infinitesimal Conform al Transformations o f IRn and 5 " 78 1.1. B T h e M öbiu s G r o u p ... 81

1.1.0 T h e Structure Equations o f the M öbiu s G r o u p ... 91

1.2 Subm anifolds o f S " ... 96

1.2. A Zeroth-O rder G'-Frame Fields A lon g / ... 97

1.2. B F irst-O rder G'-Frame Fields A lo n g / ... 98

1.2. C Second-O rder G'-Frame Fields A lon g / ...103

1 .2 D T h e Generalised W eyl Tensor and Conformally F la t Sub­ m anifolds ... 110

1.3 Relation w ith Riemannian G eom etry o f Submanifolds o f the Eu­ clidean S p a c e ... 112

Appendix I 125 2 V A R IA T IO N A L PR O BLEM S IN C O N F O R M A L G E O M E T R Y 127 2.1 Introduction: T h e W illm ore Functional ... 127

2.1.

A

T h e Riemannian C a s e ...127

2.1. B Conform al Interpretation with Further Generalisations . . . 130

2.2 The Euler-Lagrange Equation for the W illm ore Functional W . . . 135 2.3 The Conform al Gauss M a p ...153 2.4 A Conform al Bernstein-type T h e o r e m ...161

Appendix II 168

3 TH E SE C O N D V A R IA TIO N FOR W ILL M O R E SU R FACES OF

A SPACE F O R M 170

Part I

Chapter 0. Ce aérai Remark,

aad

Notai bas 2

Chapter 0

GENERAL REM ARKS AND

NOTATIONS

L et ( N n, h ) denote tw o sm ooth Riemanni&n m anifolds o f dimension m,

n , equipped with their respective Levi-C ivita connections V and V \

I f 4> : M —► N is a C ’ -rnap, then <frlT N —» M denotes the pull-back o f T N by

4», i.e. the C *-vector bundle w ith fibre at * G M given b y T ^ a)N . T h e differential

d<f> o f ^ is a G l- 1-form on M with values in <jrlT N . 4>~lT N has a Riemannian

metric in duced b y the metric h o f T N . Let V * denote the in duced connection on 4>~*TN, i.e. V * is the unique linear conn ection on 4>~lT N such that for each sm ooth section Z o f T N and x 6 M , X 6 TaM

( o .i )

T h e fir$t fundam ental fo r m o f 0 is the sem i-definite 2-covariant tensor field 4>*k. T h e second fundamental fo r m o f 4» is the section V d ^ o f the v ector bundle G * T * M ®

+ ~ lT N -> M given by

v ^ ( x , y ) = v j r ' c w n ) - M V * r ) ,

where X , Y are sm ooth v ector fields on M . T h e tension field o f ^ is the section o f 4>~lT N given by

Tt = t r » c e , ( V ^ ) .

4» is said t o be harmonic, if it has vanishing tension field. The m ap ^ is said to

4>-Chapter a Oemerml Remark* and N d i l b u 3

L e t U C A f, 0 C Stm b e open sets and x : U - * 0 be a m ap that defines a co-ordin a te system. Using the index range • ,/, & , . . . € { l , . . . , m } and writing lo c a lly the metric g on U as g (x ) = g,,dx'dxJ (here we use the index-summation

\§ij\i b y |^| the determ inant o f [ f ¡j], and by the Christoffel symbols of the L evi-C ev ita connection o f A f , we have the standard expressions

I f X = X k£ f is a sm ooth vector field on A f and * = ( * ) e x€ A f, we

have th e following formulae

I f « : A f —* IR is a (^‘ -fu n ction , then the fradienf o f « on U is given by

= <°-5>

I f A f is oriented and x is an orientation-preserving chart, then the volume element o f ( A f , g) is given by dVt = \f\g\dxl A . . . A dx*.

Let V C AT, O' c IRn be op en sets and g : V —* O ' be a co-ordinate system on AT. T h e n , using the in dex range or,/?,. . . € ( l , . . . , n ) , we have, on V , * ( » ) =

h apd g ° d y l>. Denoting b y (&"*] the inverse m atrix o f |A*^], b y *be Christoff el sy m b ols o f the L evi-O ivita connection o f AT, the first and second fundamental form a o f d : A f -* AT on U are given by (assuming that + (U ) C V )

con ven tion ), that is, ^ , and denoting by [ f ,J] the inverse matrix o f

d iv(A -) = ~ X ‘ + X ’ * n (0.3)

(0.4)

Chapter 0. General Remark

»

and Notai krai 4

and the tension field o f ^ by

( * V 7 iwp» W , / y ,

Vd**ô»> ,J a * * a * a * ) '

(

0

.

6

)

Thns harmonic m aps are locally solutions o f a system o f second-order semi-linear elliptic partial differential equations. From regularity th eory o f solutions o f elliptic equations we know t h a t C7*-harmonic maps o f sm ooth Riemannian manifolds are sm ooth (M o/6 6 ). In particular, totally geodesic C ’ -m a ps are sm ooth. Such maps carry geodesics o f M t o geodesics o f N .

N ote: if AT = Ml, E q. (0 .6 ) takes the following form

A * = (0.7)

Now assume tha t -* ( J V , h ) is an isom etric im mersion, i.e. 9 — 4*h.

T hen the mean curvature H of 4> is exactly

Let V —► M denote th e normal bundle o f T hen 4 ~ l T N = dj>(TM ) © V , where the direct sum is an orth ogon a l one. The second fundamental form is a sec­ tion o f Q * T * M ® V a n d i f is a section o f V .

I f Z is a section o f f ~ l T N , we will denote b y Z T and Z1 the orthogonal projec­ tions o f Z on the v e c t o r bundles d ^ (T M ) and V , respectively. V has an induced Riemannian metric fr o m the one o f T h e in duced connection on V is given by

viz=(vi“z)1

for each C -s e c t io n Z o f V and X € TmM , x G M .

4 is said to be a m in im a / immersion, if H = 0. T h a t is, 4 is minimal, if and only if ^ is harmonic.

4k is said to have con vtan t mean eurvo/ure , if the norm ||lfj| o f i f in V (which is

equal to the norm in + ~ lT N ) is constant.

If ^ is an isometric im m ersion o f class (7*, then 4> is said to have parallel mean

curvature , if H is a parallel (/"-section o f V , i.e.

Chapter 0. Geaerml Remarka and Not air,as 5

Since V* € M and X € T ,M y d||fT||J(X) = 2 i f ^ has parallel mean curvature, then it also has con stan t mean curvature. For n = m + 1 the converse is also true.

Given an isometric immersion in to a Euclidean space tjt: (M m,4 * h ) —» (IRn, h), the corresponding Gauss map o f 7^ : (M,4>*h) —* G '(n ,m ), w here G (n ,m ) is the Grassmannian manifold o f m -spa ces through the origin in JR", is given by 7 + (x) — d4>*(TxM ) . Considering ( ? ( » , m ) with its usual Riemannian structure (see e.g. Ref. (K o-N o/6 9 ]), we have the follow in g relation between the m ean curvature

H o f rj> and the tension field Tl4 o f -y* due to Ruh and Vilms [R u -V i/7 0 ] (see also R ef. [E e-Le/83])

T14 = m V1 H .

This equality means the following:

V * € M Tl4(x ) € m ) a n d , using the canonical identification o f

ri(WG(», m) =* (->,(*))• * (l,(* ))x =

® ( # , ( r ^ ) ) ‘ ,

we have

r „ ( x ) ( d ^ ( X ) ) = m V i H „ h V X e T .M . Hence, ^ has parallel mean curvature, iff 7^ is a harmonic map.

O n the vector bundles Q * T * M ® V , i.e. o n tensor prod­ ucts o f Riemannian v ector bundles, w e will em ploy the usual in duced Riemannian m etrics which at each fibre are the H ilbert-Schm idt inner products. In general, if

( : W —» M is a v ector bundle over a manifold A /, then C * (W ) den otes the v ector

space o f C k-sections o f W .

Note th a t we are using the follow in g sign for the curvature ten sor o f (A /, g)

« " ( X ,

Y ) Z

=

- V x V y Z

+

V y V XZ

+ V

\X,y\Z

and that, if P = |e( , «*] is a plane o f TmM , where etl e% is an orthonorm al basis o f P , the sectional curvature o f (M , g ) o f the plane P is given by

X , ( P ) = ... ... . = .

Oku pier 0. (ieaerml Remark» aad Nolalioaa 6

Let <j>: M m —* (N , k ) be an immersion o f an m -m anifold M into a Riemannian m anifold (AT,/k). For each oriented com pact d o m a in D C M (and we will denote by D the interior o f ZJ, that is, Z? = D U OD) th e volume o f 4» on D is given by

VoW = jD

mvv

»,

where dV^h is the volume element o f (D ,^ * k ).

Let (^*)*e(—«,«) be a smooth variation o f 4> such th a t the vector variation W = ^ | |=0 € G °°(+ ~ lT N ) has com pact support in D . Then it is well-known that

| v i > W . ) L . = - ¡ D m ( H t , W ) k iV r l . ,

where H+ is the m ean curvature o f

4>-T h at is, the Euler-Lagrange equation o f this variational problem reads H+ = 0, i.e. the critical points o f Vd are the minimal im m ersions.

I f ^ is a critical point o f V o, then (see Refs. [S i/6 8 ], [Sp/79]) the H ettia n of Vo at

4» satisfies

= j £ v „ ( * , ) | , _ t = f o ( j t { W ^ W ^ ) k d V ,.h. (0.8)

Here

/ ( ( i f 1 ) = - 8‘ » r l - A (Wx ) - ( B i c c i , ( » ' x ) ) - L

with R icci^ ( W x ) , = E 6 D , where X , ...X „ is an orthonorm al basis o f (7 »A f,p *A ), R N is th e curvature tensor o f (JV,h), ( ) ‘ denotes the orthogonal projection o f ¿ ~ l T N o n to the normal bundle V o f 0. A is the element o f C ~ ( ® V ® V ) given by A , ( W f ) = ] T ( V d * . ( X „ X / ),tV ,1 ) J V d ^ ( J f , , X y ) , and where denotes the Laplacian in the n o rm a l bundle:

= t V 1,Wx(X„X,) = f

- V i x w ;

1-1 1-1

(assuming that the X,- are extended as local s ection s o f T M defined on a neigh­ bourhood o f x , constituting a local frame o f M ) .

Note that we are using the opposite sign o f th e Laplacian o f Eells and Lemaire [Ee-Le/83] for sections o f Riemannian vector b u n dles, and the sign o f the Lapla­ cian o f functions adopted by Chavel (C h a/8 4j.

Chapter 0. General Remark* and Notations 7

all W e 1T N )\ {0 ) with com p act support contained in D .

The differential operator J+ : G °°(V ) —» C ° ° (V ) is the J a c o b i operator and is L ’ -selfadjoint strongly elliptic [Si/68]. A section W in G ° ° ( V ) is said to be a Jacobi field on D , if /¿(V F) = 0 on D . If Z is a Killing v ector field o n (N ,h ), that is, Z is a v e c to r field on N such that the Lie derivative L z h o f h a lo n g Z is zero, then (^ _ IZ ) X is a Jacobi field on D .

If 0 is an im mersion o f a hypersurface M m in to (j\Tm+l,fc), t h e n Eq. (0.8) is simpler. Let v denote a unit normal t o <t> on D . Then W x = u u w it h tt € C ™ (D ), th a t is, u is a sm ooth function on D with com p act support c o n ta in e d in D . In this case E q . (0 .8 ) reduces to

= Jd . ( - A . - (B + || V ^ H X r f V , . , (0.9)

= / D ( i ? . r - ( * + i v * n . ) * v . ,

where R s = R icci* ( i / „ ux) = ^ ( d ^ * ( X , ) , l / , ) d ^ , ( X , ) , I t is well-known [Fi-Sch/80] (L em m a 1, Th. lj^ [Si/68] [Sm /65] that <ft is strictly volum e-stable on

D , iff there are n o Jacobi fields defined in a subdomain D ’ c .D w h ic h are zero on *

arr.

If w e have a map 4> : -* (N n,h ) between two R iem an n ia n manifolds,

for each co m p a c t oriented domain C c M the energy o f ^ on D is given by

a . w - l / , w r w „

where dV, is th e volume element o f { D ,g ). If $ is an isom etric immersion, then Ed{4>) —

It is well-known tha t tft is a critical point o f E D, iff 0 is a harm onic m a p. If ^ is a critical point o f Ed, then, for a variation (^t)(€(_M) o f such th a t W = ^ A|t=0 G

G°°(4>~iT N ) has com pact support contained in D ,

= ( ( - « - R k c i " ( » ' ) ,

= (0.10)

Chapter

a

Central He,narks and Notations 8

A harmonic map «ft is said t o be energg-ttttble , if, fo r every oriented com pact d o ­ main Z7 c M and all W € C ° ° (4 ~ lT N ) w ith com p act support in D , Hes&ED(^ )(iy , W ) >

0.

From Eq. (0.10) it follow s obviously that, i f (TV, h) has non-positive sectional c u r ­ vatures, any harmonic m ap «ft: (M ,g ) —► (TV, h) is energy-stable.

F or a minimal isom etric im m ersion «ft : (Ai,«ft*h) —» (TV, h ), the following relation between HessVD( * ) and H ess-E oi*) holds, for W € 0 ? ( V ) (see R ef. (Fe/85)):

= H

css

K

d

(^)(

iv

,

w

) + i j D |(V*’ V ) T||’« v . .

O n a Riemannian m anifold can be defined some very im portant constants, v iz. Gheeger, isoperim etric, and Sobolev constants. These constants may provide estimates o f eigenvalues and eigenfunctions for the Laplacian operator on dom ains o f M (relative to the D irichlet problem ). One can find an extensive study on these constants in Refs. [C h a /8 4 ], [B e-G a-M a /7 1 ]. In this manuscript we are only g oin g t o use the Gheeger constant, the definition o f which we give here.

Let (A fm,g ) be a n on -com pa ct oriented Riemannian m anifold w ith dimension m > 2, and possibly having a boundary. The Gheeger com tant o f M is the non-negative number

where D ranges over all op en submanifolds o f M w ith com p act closure in M a n d sm ooth boundary, V ( D ) is th e volum e o f D , and A (d D ) is the area o f the bou n da ry o f D .

D ue to a result o f Yau [Y a /7 5 j (see also R ef. [G ha/84], Theorem 5, page 9 8 ), in the definition o f f ) ( M ) it suffices to let D range over open submanifolds o f

M that are conn ected. W e note that, if M were com p act (w ith out bounda ry),

the constant (A /) defined as above would be zero. In fact, there is a different definition o f the Gheeger constant for a com p act manifold (see Ref. (G ha/84]), b u t we are not going to need it.

The simplest exam ple o f com plete n on-com pact Riemannian manifolds w ith Gheeger constant equal t o zero are the timple Riemannian manifolds, i.e. the R ie­ mannian manifolds ( M m,g ) such that there exists a diffeomorphism «ft: (TV/,f) —* (2Rm, < , > ) onto JRm satisfying Ag < , > < pg for some positive constants A ,p .

Chapter 0. General Remarks mad Notations

with positive Cheeger constant, as fo r exam ple the m -hyperbolic space. In fa ct, if

{ M m,g ) is a com plete sim ply conn ected Riemannian m-dimensional manifold with

sectional curvatures bounded from above b y K , where K is a negative constant, then (Ya/75)

( m - l J v / T j r . . (O .n )

This result was obtained by using B ish op 's comparison theorem to arrive at A r > (m — 1 )y/—K , where r is the distance fu n ction to a fixed point in (A /, g), integrating A r , and using Stokes’ theorem.

A nother way o f estimating ( A /) is the following inequality due to Gheeger (see e.g. Ref. [C ha/84], theorem 3, page 95)

H D ) > i ^ ’ ( D ), V£> c AM onuun ,

where A (D ) is the first eigenvalue for the D irichlet problem in the domain D . Using this fact and an estimate o f the low est Dirichlet eigenvalue o f the geodesic disk o f radius 6 in the m -h yperbolic sp ace U m o f constant sectional curvature

K — —1, one ca n see that = m — 1 (see Ref. [C ha/84], page 96), so

Chapter I, Set I torn I. latrodaeiioa 10

Chapter 1

THE M EAN CURVATURE OF A

GRAPH

1.1

In trod u ction

In 1955 Heinz [H e/55] proved that, if * = z (x ,jr ) is a surface o f JR* defined for

z * + y* < R* with mean curvature satisfying \\H\\ > a > 0, then R ^ J. Thus, in

particular, if x is defined in all JR*, then inf||lf|| = 0, w hich implies that, if z has constant mean curvature, z must be a minimal surface o f IR*. In 1965 Chern (see Ref. [C h/6 5 ], Cor. o f T h . 1) and, independently, Fla nders (Fla/60) obtained the same result for hypersurfaces o f 2RB+I defined by the equation z — z ( x i,

One can form ulate a generalisation o f the above problem as follows: Given two sm ooth Riemannian m anifolds (A /, g ), ( N , h ) and a sm ooth map / :

M - * AT, the graph o f / , 1 / = { ( * , / ( * ) ) : x € A / } , is a »/»-submanifold o f the

product M x N o f co-dimension n. W e take on AI x N the Riemannian metric product g x h and on 1/ the in duced one.

Q u e s t io n (E e lls ) A itu m e that T/ hat parallel m ean curvature. D uet thii imply

Tf to he a minimal tuhmanifold o f A / x N 9

Chapter J, Seetloa I. latrodaelkm 11

submersion, a harmonic morphism, or when n = 1.

In the general case we are going to impose a cond ition on the Riem annian manifold (A f, g) that positively answers the above question (see Cor. 1.1.2). M oreover, we w ill also show that the absence o f this cond ition conjures up coun ter-exam ples (see T h . 1.2).

Let us consider 1/ as an embedding

I> : A f - ( M x J V . j x k )

X —

(*,/(*)).

S o we have two Riemannian m etrics on A f, viz. g and the one in du ced b y I / ,

r /(fx * ) = f + A ,

which makes 1/ : (A /, g + f*h ) —* (A f x N ,g x h ) an isometric im m ersion. L et V and V * denote the L evi-C ivita conn ections on (M ,g ) a n d (A f, g + /*/»), respectively. Let V be the normal bundle o f 1 / in T f lT ( M x AT) = T A f x f ~ lT N and V ’ rfl) € C ° ° ( 0 * T * M <g> V ) be the second fundamental form o f th e immersion I / . The mean curvature o f I ) is the section

o f V . Let € O m( & T 9M ® f ‘ lT N ) be the second fundam en tal form of the map / and 7 ) its tension field, when A f is considered with the m etric g. We denote by and V r' the induced connections on f ~ lT N and i y ^ T f A f x AT), respectively, and V 1 denotes the connection on the normal bundle V . L et ( , ) x a n d ( , ) T denote the orthogonal projections o l T M x / ~ lT N on V a n d o n dT/(TAi), respectively, relative to the metric g x h.

In general, there is no natural way to relate the Levi-Civita con n ection s V and V * o f resp. (A f, g) and (A f, g + f* h ), but we have the following relation am ong the connections V , V * , and V r/ s

i f X € 0 ° ° ( T M ) , U e 0 ° ° ( f l T N ) , then (X , U ) given by (X , U ) . = ( X . , U .) , Vx € A f , is an element o f G 0D( I j ‘ , T (A f X AT)) and we have

Vy1' (x, u) = (

v f u ) , w

e

.

(i.i)

Oka pier I, Seetioa I. Tke dementi Came

_{1 2}

1.2

T h e G eneral Case

Next we are going to derive an expression for the mean curvature o f I ) and its covariant derivative in V .

Let be a local orthonorm al frame o f (A /, g). Defining

i ; ~ + m X , ) , < M X j ) ) t , Vi, j E {1 ... m } (1 .2)

and denoting b y the inverse o f the m atrix tee have

m B = f ; ¡ ‘' V ’ O H X i . X j ) .

Let (7 ,')i£ i£ m b e a local orthonorm al frame o f (A f, p + /* A ). Then, (X , , d f ( X , ))i< ,< m is a local orthonorm al frame o f d T t(T M ). N ext we define the following sections

w e c “ ( / - * r j v ) , z e c ~ ( r A f )

W = trace,,+/**,(V c(f) (1.3)

( i .4 ) i j - i

We note tha t Z is well defined over all M and th a t another way t o write Z is

z = f ; ( i v 1d f ( y , ) ) i x . ( i . s )

Then we can form ulate the following lemma:

Lemma 1.1 VX, Y G C7°°(rM)

(.') V'dI>(X,y) = (0,V<tf{X,Y))X

( « ) m f f = ( - Z . V V - d f ( Z ) ) = (O .IV )1

(•••) "> V x' ‘ f f = ( o , v i " V - v j/ (x,z) ) - ( V x z . d f ( V x Z ) )

mVjfff = (0, V jf V - V<V(Jf,Z))x

Prtw/. Using E q . (1.1) we have

V d T yfx.r) = V.T(dr/ (y '))-d r / (V ;y )

= v 7 '( y , d f ( y ) ) - ( v ; y , ^ ( V ; y ) )

= ( v x y , v f ‘ w ( Y ) ) ) - ( v ; y , d f ( v ; y ) )

= (v * y - v ;y , Vdf (X, y ) + df( Vxy - v ; y ) )

Chapter I, Seelioa I. The Ceaera/ Caae

13

Since V d I ) ( X , y ) € C7°°(V), we get (i). Thus, we have

mH

= £ #"vvr,(x„xi) = (o, £

l’li™l I ja l

= (0,tr*ce(,+/.,l(ViV))i

= (0 .W )-1- = (0 ,tV ) - (0 ,»V )T .

Since

(0,WT

= E " , ( ( 0 , (V ), ( X „ < t f ( X ,) ) ) ^ ( X „ * ( X , ) ) =

E” . (tV,

^ f(X | ))4 • ( X „ # ( X , ) ) ,

»•* = (0,(V) - ( £ (W'.tffX,)). X , £ (w, <f(X,))t 4T(X,))

= ( 0 , t K ) - ( z ' V ( Z ) ) ,

which gives (ii).

Finally, differentiating the latter expression and using Eq. (1.1) we obtain

m V T . t r = ( O . V i V ) - ( V x Z , V f \ j f { Z ) ) )

= (o, v f V) - <vxz,

v<mx, z)

+.

if(Vxz))

= (o,V f ‘ w - W (X ,Z)) - (VXZ,^(VXZ)) . V

The follow ing lemma will often be used.

L e m m a 1 .3 Let x 6 M , X € TJV /, and * e T/{m)N . Then ( X ,0 ) , ( 0 ,* ) € T ,M x r /(,,JV and

(i) * = 0 iff (0 ,* ) x = 0 (ti) ( X , 0 ) € V . iff X = 0 .

Proof. A t the point x we have

(0,

= (o,») - (0, C)T = (o,») - £ <(0,.), (X„ -(MX,)))^ (X„ <tr.(X,))

= ( - E ( « , « r . W ) ) , T < . * - £ < * , <tr.(x,)),4r.(x,)).

(i.«>

If (0 ,* ) x = 0, then the first com ponent o f the vector in E q. (1.0) is also aero. Therefore, since (X ,)i£ .£ m is a basis o f 7*.M, = 0, Vi € ( l , . . . , m ) , and the v ecto r in Eq. (1.6) becom es

Chapter i, SteOoa

I.

The Oemerml

Case

14

T h at is, z = 0 and (i) is proved. Now we prove (ii):

If ( X ,0 ) € V „ then V F € T ,M < ( X ,0 ) ,( Y , 4 f .( Y ) ) ) txk = 0, so ( X . V ) , = 0. Hence X = 0 .

In R ef. [Ee/79] it was pointed out that T/ is m inim al, iff

id : (A /,g + f h ) - (A /, g) f : (A /, g + fVh) - (1ST, h ) (1.7)

are b oth harmonic maps.

In fa ct, since I ) = (id, / ) : (AT, g + f*h) -* (A f x N ,g x h ), tnH = (T.*j, T /) , where are the tension fields o f the maps id and / in E q . (1.7).

The system (1.7) can b e reduced to an equivalent equation.

Propoaition 1.1 The follow ing itatem en tt art equivalent:

(i) I/ is minimal,

(••) / : (A f,g + f*h ) -* (AT, h ) i t harmonic, (i ll ) W = (ln K «)(f+/.,](V < tf) = 0.

A lto, Tf i t a totally geode tie tub manifold o f M x AT , iff / : (M ,g ) —* ( N ,h ) it a

totally geodetic map.

Proof. From Lemma 1.1 (ii) — ( - Z , W — d f { Z ) ) % hanca —Z ■« and IV — d f ( Z ) = T f. Therefore, i f (i) holds, then (ii) a n d (iii) obviously hold. I f (ii) holds, that is T f = 0, then m H = ( 7 j j ,0 ) . S o , as H € C ° ° (V ) and from Lem m a 1.2 (ii), H = 0.

If (iii) h olds, then m H = ~ ( Z ,d f ( Z ) ) e V n dTf ( T M ) . S o H = 0. The last statement follow s im m ediately from Lem m as l . l ( i ) and 1 .2 (i). 9

T o prove the main theorem o f part one o f this w ork we recall the following formula (see Ref. (E e-L e/78], page 9):

Given a m ap ^ : ( P ,,g i) —* ( P , , gt ) between Riem annian manifolds, w e have

= + , (i.8 )

where dj> ■ T+ is the v ector field o f T P t given by

Chapter 1, Section i. The Cenemi Cane 15

and where ( , ) is the in duced Riem&nnian m etric in the v ector bundle ® T *PX ®

In particular, if 4> is an isometric im m ersion, then, since T+ is orthogonal to d<t>(TPt), E q. (1.8) b ecom es

T h is formula can easily be com puted directly, to o .

T h e o r e m 1 .1 A ttu m e that 1 / hat parallel m ean curvature. L et e = ||ff||#)<A (c it

a eonitant). Then, f o r each oriented eom pact domain D C M , toe h a te

j_A(aD)

- m V { D ) '

where V ( D ) i t the volume o f D and A (d D ) i t the area o f dD, relative to the metric

9-P roof. From Lem m a 1.1 (iii) we have V X € G ° ° (T M )

L et xo G M and X j , . . . , X m be a local orthonorm al frame o f defined in a neighbourhood o f x0 and satisfying V X ,( x 0) = 0, Vi = 1 , . . . , m . Such frames can be constructed using parallel transport in (A /, g ).

* ' T P „ that it, V * e M , ( i * , V r \ ) (* ) = E n . where « 1 ,. . . , ep is an orthonorm al basis o f T*P\.

(1.9)

0 = m^Jx H = (0, V C ' w - V d fiX .Z ))^ ,

hence, from L em m a 1.2 (i),

v ¡ ¡ " w = \ j f ( x ,z ) .

From Lem m a 1.1 (Ui)

™ Vt/ 'h = ~ { V x Z , 4 f ( V x Z ) ) . ( U O )

From the latter equation we can prove now tha t

Chapter l, Section

J.

The General Cate 16

T hen, Z = ly V ,d f(X i))h X j in a neighbourhood o f *

0-Since V Y ,( * o ) = 0, we have at the point x0

Y v . z = £ V * , ( i ‘ '< W ,< V (A 1) ) , A , )

**= I

= £ </(«*’ < W ,< < r (A ,,))J (A ,)A , ,

hj=l

so V», j

{ ( V x , Z , 4 f ( V x , Z ) ) , ( A j . X K A y ) ) ) ^ =

= £ r , A i , ’ < y v ,4 n x > ) ) > ) i x i) ,

**= i

a n d, therefore, from Eq. (1.10)

. n ( v r'" i r , f l > ) ( x o ) = E ^ ' ' (yJ T * , * > ( * / ) )

'

'

i> l

'

'#**

= £ - f “ < ( V * z , <V(VX. Z ) ) , (J O ,< r ( A ,))>

ij-l

’

• •A*** 1

= £ - M ( i ‘ ' R « U K )

= £ - « '( » * ’ W ^ A » ) ) , ) „ ( A . ) .

• »*■1

Since E E .I 0*' (W ,d /( X * ))A = ( Z ,X j ) f , Vi = 1 , . . . ,m in a neighbourh ood o f x 0,

m ( V > _‘ A ,d r / ) ( x ll) = £ -< i( < Z ,X ,)( ) ^ ( A . ) = £ -

(VX,Z,

A , ) ( (x„) = - d i v , ( Z ) ( * . )

and we have proved Eq. (1.11). O n the other hand, from E q. (1.9) we have

( V ' / 'A .r f T ,) = - m i l i a r = —me* .

So Eq. (1.11) gives

Chapter 1, Section ». The General Case 17

L et 5 C M be an oriented com p act dom ain and d,Vt ,d A t den ote the respective volu m e elements of D and d D relative to the m etric g. A p p lyin g Stokes’ theorem w e get

w here ft is the outward unit normal o f dD.

F rom the Schwarz inequality | (Z ,ft)f \ < ||Z||,||ff||, — ||Z||, and Lem m a l . l ( i i ) , we

m'c'V(D) <

| (Z, ft)f |dA, < J mcdAt = mcA{dD) ,

^ 1 A (d D ) n

so c < —

777

- . .

m V ( D )

C o r o l l a r y 1 .1 .1 I f (M , g ) i t an oriented non-com pact R iem annian manifold and

f : M —♦ N it a smooth m ap such that 1/ h a t parallel mean cu rtature H , then

C o r o l l a r y 1 .1 .2 I f (A /, g ) i t an oriented, compact (without boundary) m anifold

o r an oriented non-com pact Riemannian m anifold with G heeger constant equal to

zero (tee Gh. 0 f o r definition), then fo r any Riemannian m anifold (N ,h ) and any

m ap / : (M ,g ) -♦ (AT, A ), i f the graph 1} : ( M ,g + f* h ) - » ( M x N ,g x h ) it an im ­

m ersion with parallel mean curtature, it i t in fa ct a minimal submanifold o f M x N .

In Chapter 0 we recalled th a t, if (A /, g) is a simply con n ected Riemannian m - dim ensional m anifold with sectional curvatures bounded fr o m above by

K ,

where

K

is a negative constant, th en

P)

(AT) £ (m -

\ )y / - K ,

a n d that, i f

M

is the m -hyperbolic space, P) ( A /) = m - 1. Therefore, in such cases C or. 1.1.2 ca n n ot b e applied. M oreover, we will give next an explicit example w hich shows tha t the con d ition on the Cheeger constant o f (M ,g ) is a fundamental criterion for a graph w ith parallel mean curvature to be minimal.

r n V V ( B ) = J ^ m 'c ’ dV, = ^ d iv ,(Z )< < V ',

ob ta in

m . = m ||if|U = ||( -

Z ,W

-

4 n Z ))\ U

> ||Z||, .

Hence

-Chaplet I, Seetiom

J.

The Gemetal

Caw-

18

T h e o r e m 1 .2 G on tid er the 2-dim eniional hyperbolic »pace (H * ,g ), where H * it

the unit open ditk o f 1R1 with centre at the origin and g it the Riem annian m etric

on H * giten by

t . _ W

'

(i -1*1’ )*

The fu n c tion f : H * —* IR given by

(1.13)

/(*) =

Jt

^ i(c °« k (r )-

1

) * ,

where r (x ) = log (| 4 ^ | ) »* the dittance fu n c tio n from the origin in H * , it tm ooth

on all H *, and 1 / C H * x IR hat conitant m ean curvature ||iT|| =

Proof. It follow s from Lem m a (1.3), to b e given and proved in the next section,

that we on ly have to v erify i f / satisfies the equation d iv, ( = First we calculate the Christoffel symbols o f the m -hyperbolic space (H m, g ), where

g is given b y E q . (1 .1 3 ). Defining the id en tity map x : H m —* IRm as a c o ­

ordinate system , we have ^ where ¿ i , . . . , e m is the canonic basis o f 2Rm. Let g,j = g (e,-,e j) = (1_ |*|i)i and \g,}] b e the inverse m atrix o f [flftJ], th a t is,

gij _ T h e n , using Eq. (0 .2), we ob tain 2

r j = |x |, {6kjZ, + bikXj - 6,jXk) . Now we prove th a t / is sm ooth.

V* 6 f f * \ { 0 } , « e T „H * = IR*, we have dfs (u ) = ^ j( c o s h ( r ( x ) ) — l)d r M(u ). N ote that c o s h (r ( * )) = and that d r ,(u ) = jzy, [« So

2 < x, « > <r- ( “ ) = ( i - i-i» )»/» • Now we show th a t §£ i(0 ) = 0 for i = 1 ,2.

(1.14)

lim ! / ( * « . ) - / ( Q ) I N ettmak-Ulhl) / T “ " I

i * / .

v»(

1)A

Since lim ,_ = 1 and tanh 1 : ( —1 ,1 ) —* ( —o o ,+ o o ) is an in creasing function, we have V3 > 0 ,3 c > 0 such th a t, VA : 0 < |A| < c ,

and, Vt € [ 0 ,2 ta n h ~ l (|A|)], y/| (cosht — 1) < S. Hence,

<

1

+

6

N /■! I»nh_1 (|A|) IJ

Chapter

1,

Sec Hob >. The Geaeral

C u e

19

So we have proved that E q . (1 .1 4 ) also holds for * = 0, which p roves the sm ooth­ ness o f / on all H *.

The vector fields « ,( * ) = — i — 1 ,. . . , m form an orthonorm al frame o f

R e m a r k 1 .1 A s a consequen ce o f C or. 1.1.2, P rop. 1.1, and H o p f’s maximum principle (see for example R e f. [C h a /8 4 ]), i f M is an oriented co m p a c t manifold,

N = jR " , and I ) has parallel m ean curvature, then / is a con sta n t map.

R e m a r k 1 .3 In Sec. 1 we presented the result o f C hem [C h /6 5 ] on the mean curvature o f a graph as a sta rtin g p oin t for the main theorem o f th is section. This result was a corollary o f a t h e o re m in his paper quoted above, w h ic h we reproduce here:

T h e o r e m ( C h e r n ) Let P be a compact piece o f an orien ted hypenurface o f dim ention m with im ooth boundary d P which i t im m erted in a Euclidean tpace

o f dim ention m + 1. Suppote the mean curvature o x > c > 0 . L et a be a fixed

unit vector which m aket an angle < f with all normal» o f M . T h en mcV. < L .,

where V, it the volume o f th e orthogonal projection o f P and L m that o f SP in the

hyperplane perpendicular to a . I f M it defined by the equation z — F ( * , , . . . , x m),

f o r ** + . . . + < R , then c R < 1.

The above case seems, at first sigh t, much more general than a gra p h , but, in fact, it is essentially the same, as w e are going to explain in detail.

The condition . . Let a be a fixed unit vector which maket an an gle < J with all

Finally .e « I d . « , div, ( ^ r f o y ).

HT»L _____ a__C _1 • l \ _ 1—1*1* _

So

v / . =

E Z , « . ( W

=

*»d »V/K» =

Chapter I, See Hob I. The Gemeral Case

2 0

normalt o f P . . . ” means the following:

Let us assume that the angles are < D e n o te b y $ : P " 1 —► 2Rfn+I the immersion o f P into the m + 1- dimensional E uclidean space, and le t 4> : P —» [a]x ^ JRm denote the com position o f $ with the orth og on a l projection o f JRm+l onto [a]x . That is, = $ ( x ) — ($ (x ), a) a, Vx e P . T h en 4> is also an immersion o f P , as follows straightforwardly from our assum ption concerning the angles.

V■ and L a in C hem ’s theorem are resp. the volum e and area o f P and OP relative

to the metric induced by the immersion ^ o f P in to [a]x £* IRm. Writing now 2Rm+I = |«JX x («], then

* : P - R m+l = W A x [a] is given by

$ (x ) = (<£(x), < $ ( x ) ,a > a ) cs ( ^ ( x ) ,< $ ( x ) , a > ) , since ^ ( x ) + < $ ( x ) , a > a = ♦(*)■

Thus $ can be written as $ (x ) = ( 0 ( x ) , / ( x ) ) , where / : P m —» 1R cs [a] is a smooth m ap and 4» : P m —* JRm c* [o]x is an im mersion. W e can consider $ as a parametritation o f a graph, where the first com ponent o f $ is the isometric immersion <f> : P m —» IRm instead o f the id e n tity m ap, which is the case o f a graph.

In the same spirit, we can also im prove o u r main theorem (1.1) for the case o f a parametritation o f a graph:

Let (N * ,h ) b e sm ooth Riemannian m anifolds and P m an m-dimensional

manifold. Let • = (^ , / ) : (P , 4>*g + f * h ) —• (M x JV, g x h ) b e an isometric immersion with com ponents 4> and / , such th a t 4> : ( P m,4>*g) —* (AI m,g ) is an isometric immersion and / : P —* N is a m a p . Let H b e the m ean curvature o f the isometric immersion $ . Then Th. 1.1 ca n b e reformulated as follows: Theorem 1.1* I f $ h a t parallel mean curvature, then, f o r each compact oriented domain Tf c P , we have

e < 1 A (O D )

' m V(D) 9

where c — ||£f||#xA (contta nt), V ( D ) and A ( d D ) are retp. the volume o f D and the

a n a o f OD relative to the metric

Chapter I,

Stri torn t. CoDImtaaioa Oat

21

1 .3

C o-D im ension One

I f th e graph 1 / is a hypersurface o f A f X N , th a t is, N is o f dimension on e , we

can obtain an estimate for the infimum o f the norm of the m ean curvature o f I ) , w ith ou t needing to im pose the assumption o f 1 / having parallel mean curvature, as in the general case.

L e t us suppose that N is oriented and o f dimension o n e . Let Y be a unit v e cto r field defined on a ll (j\T, h ). Define u : = ^ 1 + ||<{f||*, w here ||4f|| is the n orm o f d f in Riemannian vector bundle & T * M ® f ~ l T N . D en ote b y V / the sm o o th section o f T M given b y < V / „ u)f = (4 f,(u ), Ya) k , V* € M , « € T J lf. T h u s, ll^ /ll# = ll<V|| v — 'V / , F ) is a unit normal o f I ).

In th is case i t is easy to derive an expression for the m atrix (here we use the same notations as in Sec. 1.2). D enoting p, = ( d f ( X , ) , Y ) h, w e have

For M — IRm and N — IR, this expression is equal to the one ob ta in ed b y F landers (F la /6 6 ].

m B = £

<J-1 From Lem m a 1.1 (i), we have

= £ v \ f l } ( x „ x , ) - £ ¿ w V « > ( x ( , X j ) .

1=1 i w

V<n>(x„

x,)

= <(o,

Vd/(x„ xj))

= i ( Vrf/(x„ x,), y)^.

H ence,

Chapter i, Section t. Co-Dimension One 22

In particular, die, ( - ^ S £ _ ) = m e (e eonctant/, iff ||if||,j = |c|.

Proof. Let xo € A / , a n d X i , . . . , X m b e a local orthonorm al frame o f ( A /, g) in a

neighbourhood o f

x0,

such that

VX,(x0)

=

0.

Then, at the point

x0,

= t ( v r ( ^ m ) , r \

= t (< H ±)(X .W (x,) + ^ V d f(x „x , ) ,

=

4f(X) +

'- I ,, y

y

= ( - ¿ £ <V*(<r,*>«r(X) + ± r,, y ^

= ( - E ¿ < v * ( x , x ) , t f ( X ) > . « i r ( x ) + z T< ■

' « j - l w / A

= ( - 4 £ < V<r(x,x,),y > . <

j h x, ) ,y

>„

d f ( X t )

+ ~T/, y

' w i j - l w

*■ in <. H , v > tHk (» o ) (from E q. (1.16)).

Let ||V<y|| denote th e norm o f Wdf in Q * T * M ® f ~ lT N .

P r o p o s it i o n 1 .2

(a ) I f D C. M it an oriented eompaet domain o f M , then

niin||£r||fX* <

V

11 ll#x* “ m

V ( D ) ’

where A (# D ) and V ( D ) are retp. the area o f dD and the volume o f D , relative

to the m etric g. In particular, i f (M , g ) it a compact manifold or n on -com p a ct with Gheeger constan t equal to zero, then inf||lT||,xA = 0.

(1 ) I f (A /, g) i t a connected, oriented, complete Riem annian m an ifold and - J O a ^ i . i n t c r a U . in ( M ,g ) , then there exist» a x € M , such th a t H „ = 0.

Chapter

I,

Seel loa t. Co-Dimension Oae

23

Proof, (a) Let e = min||lf ||. Clearly we m ay suppose e £ 0. Since D is connected,

Denoting by dVt and d A , the volum e elements o f D and otD, respectively, and by

H the outward unit normal o f dD , and applying Lem m a 1.3 and Stokes’ theorem ,

we obtain

(b ). Suppose that H* ^ 0, V * € AT. N ote that, in this case, E q . (1.16) holds on all A /.

Let 9 = where ♦ : h l T * M — is the Hodge operator . Then, dO = — divf { ^ f)d V l . Since we are supposing that ||^||f is

inte-bundles f\l T * M and Am_I T * M , ||0|| is integrable on (A /, g), i.e. 9 is an integrable (m —l)-fo r m o f (A /, g). B y applying the extended Stokes’ theorem o f Gaffney-Yau (see Ref. [Y a/76], lemma o f Sec. 1) to 9, we m ay take a sequence o f com pact do­ mains D, o f A /, such that B , C B,-+i, V», U # . = A /, and lim ,_ +00/ flj d9 = 0, that is,

Consequently, JB( ||R'||#x*dV,# = 0, VS, i.e. H = 0, wliich is a contradiction. If we suppose that < H , v > #x* is contained in | 0 ,+ oo) or in ( - o o , 0 ] , then, again, Eq. (1.16) holds on all A /, which implies I f = 0 as well. V

||ff|L = < f f „ V * € D

or _{( i - i « )}

1 * 1 . = Vi e d .

' V ( D ) <

¡ D

||*|„»«fV1 =

\jD

< H , v

>fx4

iV,\

Therefore, we can conclude from E q. (1.16) and Lem m a 1.3

Chapter I, Section 4. G rap ht of hornet rie Immenioat, Conformai Maps, . . . 24

R e m a r k l . S In Prop. 1.2(b) (i) we could only require the weaker condition

for some x 0 € M , where 2?A (x 0) is the geodesic ball o f (M ,g ) with centre x 0 and radius R . In fact, the Stokes’ theorem o f Yau still holds with this condition (see A p pend ix o f Ref. [Y a/76]).

1.4

G raphs o f Isom etric Im m ersions, C onform al

M a p s, R iem annian Subm ersions, and Har­

m on ic M orphism s

In Sec. 2 we have seen that for a m ap / : (M ,g ) —» ( N ,h ), 1 / to be minimal is in general not equivalent to / : (M , g) —* (N , h) be harm onic. However, we will treat some cases w here the equivalence does hold.

A m ap : ( P u t t ) ( A . i s ) between tw o Riemannian manifolds is said to be (weakly) conform al, if — f ? g i, where p : P i —* IR is a sm ooth map. If

dim P i > dirnPj, then 4> is constant. If p is a non-zero constant, 4> is said to be a hom othetic m ap and, in particular, an isom etric im m ersion, if p ■= 1.

If

p(.x )

/ o

V * € P i, l.e.

<f>

is an immersion, then we have the follow ing well-known relation [H o-O s/82] between 7*, the tension field o f <f>: (P i, gi) - * (P t ,g t ) and H+, th e mean curvature o f the isom etric immersion 4» : (Pu<t>*gj ) —* (P i, g*):

where m — d im (P ,) and w — V#1 lo g p . We recall that a Riemannian manifold

(M , g) is said to be (strongly) parabolic, if it a dm its no non-constant subharmonic

functions / (i.e. A / > 0) that are bounded from above.

P r o p o s i t i o n l . S Lei f : (M m,g ) - » (AT", A) he a conform ât map with f * h = A*g.

L et H be the mean cureature o fT f. Then tte hawe:

Ckaptrr

!,

Srctiom 4. Oimplm at konteirk Immenioam, Comtormml

Maps, ...

25

(•)

m f f = (0 , (1 + A, ) - , T /)'L .

In particular, i f x 0 € M , H ^ = 0, iff T/(x0) = 0. T herefore, 1 / it a minimal tubmanifold o f (M x N , g x h ), iff f : (M ,g ) —* (N ,h ) is a h a rm on ic map (and

in thit ca te, f o r m £ 2 , f it a homothetic m ap).

(b) I f m — 2, or f it an itom etric im m ertion or, m ore generally, f i t a hom othetic map, then Iy h a t parallel mean eurtature, iff Iy it minimal.

( c ) I f m ¿ 2 and Iy ha t parallel mean curtature, then

with c = ||£T|| (contta nt). Coniequently,

(i) i f (M , g ) it parabolic or i f A hat a minimum on M \ d M f o r tn > 3, then Iy it minimal.

(ii) i f (M m, g ) it com plete, connected, and oriented, and m > 3, then fo r t o l (M ,g ) < + o o Iy it minimal, and f o r v o l(M ,g ) = + o o (1 + A*)-1 £

l S ( M ,g ) , V p G ( l , + o o ) .

P roof. Since f* h = A*g, T f(g x h ) = g + f * h = (1 + A*)fl = p *g, where p : M -*

| l,+ o o ) is a sm ooth m ap. It follows from Eq. (1.17) that

m U = + (tn - 2)fi~‘ (w ,d f ( w ) ) ,

w ith Tr; the tension field o f Iy : (M ,g ) -* (A f x N ,g x h) and w ith to = V ,lo g /* . Thus, Tr/ = (Tid ,Ty) = (0 ,T y). Hence,

m H = p - *(0,Ty) + ( m - 2 )p - , (t o ,4 f(t o )) • (1-1*) So m H = ( m H ) 1- = (0 ,/*~*T/ ) J- and (a) is proved b y applying L em m a 1.2(i). I f / is harm onic, i.e. H = 0 (from (a )), and m £ 2, then Eq. (1*13) gives to = 0, that is, / is a hom othetic map (see also Ref. [E e-Le/83]).

N ext we prove (b ). I f m = 2 or / is a hom othetic m ap (i.e to = 0 ), we obtain from E q. (1.18)

mH

= p->(0,r,) .

S o, applying formula (1.1) we have, V X € C ° ° (T M ) ,

Chapter

J,

Stxiiom 4. Graph» o f hornet tie Immenioaa, Coaformal Napa, . . . 26

and

m V ^ B = ( 0 , V f ‘ (r~ , T , ) ) i .

(1.20)

H ence, applying Lem m a 1.2(i) to E q. (1.20), we conclude that

V x B = 0 iff V y (n~*Tj) = 0, which is equivalent to H = 0 due to Eq.

(1 .1 9 ). Using E q. (1.9) for tf> = I / , we get V 1* H = 0 iff H = 0 ,‘ and we have

proved (b ). In order to obtain (c ) we are first going to prove the following formula:

< »> .# (■ ))» = •»*. (1.21)

L e t us fix x 0 € M , and let X i , . . . , X m be a local orthonormal frame o f ( M ,g ) defined in a neighbourhood o f x 0 and satisfying V X , (* 0) = 0, Vi = 1 , . . . , m . A t x o we have, V», / , k € { 1 , . . . , m },

( V jf ( x „ x i),4f(>:» )),(* ,) = ( v C(<V(Xj)),inx>))k

= 4 < j n x i ) , d n x t ) > k ) m - ( v m , v # ( x „ x t ) ) k

= W ( X l) - { 4 n X i),V4HXl,X t))k .

Perform ing a cyclic perm utation on the indices * , / ,

k

we get

{■V4f(X„Xj),4f(Xk))k = tJt4 X ^ X ,)-(4f(X l),Vdf(Xl,X k)}t

(V4nxk,x ,).4 n x j))t m t,jd\‘ (Xk) - (4f(X,),V<t/{Xk,X 1))k

(Vdf(XJ,X k),Jf{X<))k = W ( X i ) - { 4 f ( X k),V4f(XJ,X l))k ,

a n d so, at the p oin t x0,

(■VJf(x„x,),dnxk))k = i {<Jlr<tt*(x,) _ t„ ix \ x k) + e ^ X j ) } .

H ence, for t = j

(Vif(XhX,),df(Xk))k = 6,kdX'(X.) - i i * ’ ( X . ) ,

( T ,.d f ( x k) ) k ( Ik ) = Y . ( y m x 1. x i ) , d n x k) ) k

H

Chapter I, Section

t

Graphs a t hornet rk Immersions, Conformal Maps,

...

27

Supposing that 1 / has parallel mean curvature we have, from E q . (1.12) in the p r o o f o f T h . 1.1,

m*e* = d iv , (Z ) ,

where Z is the v e cto r field o f A /, given in E q. (1.4). N ext we are going to prove that

m V = ~~2~ A (#*"*) on M . (1-22) Let z 0 € A / and X i t . . . , X m be a loca l orthonorm al frame o f (A f,g ) defined in a neighbourhood o f x 0 and satisfying V X ,( * 0) = 0, V» s l , . . . , m . Then,

Z = £ * 'J < W , m x . ) > . X j and W = £

•J"l ijm 1

C km pier

I,

Sec lio a 4. Grapka c f hornet rk lmmenioma. Coa formal Mapa, . . . 28

H ence, i f (M ,g ) is parabolic, p m ust be constant, and, therefore, 0 = A ( p - *) = i.e. 1 / is minimal.

For m > 3, i f A has a m inim um on M \ d M , then p~* has a m aximum on M \ d M . A s A / i - * > 0, it follow s from H o p f’s m aximum principle, applied on a bounded dom ain o f A / where th a t m axim um is attained (see e.g. R ef. [A u /8 2 ], page 96), th a t (c ) (i) holds.

N ow w e prove (c )(ii). From Eq. (1.23) we have p~* A (p _>) > 0. S o, from T h . 3 o f R ef. (Y a/76), we have either / w = + o o , Vp G ( 0 , + o o ) \ { l ) , or p is constant. Thus, if the volum e o f ( M ,g ) is finite, we conclude from 0 < p~* < 1 th a t p~* G L p( M ,g ) , Vp G (1, + o o ) . Therefore, p is constant and 1 / is minimal. L et us now suppose th a t the volum e o f (M ,g ) is infinite. I f p~* G L p(M ,g ) fo r some p G ( l , + o o ) were true, then p w ould be constant. Since p cannot be equal to zero, this would im p ly th a t the volum e o f (M , g) is finite, which is a contradiction. So, in this case, p - * £ l / ( M ,g ) , Vp G ( l , + o o ) .

R e m a r k 1 .4 P rop . 1.3(c) means th a t, if m > 3, v o l( M ,g ) = + 00, and 1 / has n on-zero parallel mean curvature, th en , Vp G (1 ,- f o o ) , ( l + P cannot be

integrable, nor have a maximum.

N ow we study the graphs o f R iem am úan submersions and harm onic morphisms. H enceforth, until the end o f this section, we assume tha t ( M ,g ) and (AT, A) are boundaryless manifolds.

Let / : (A im,g ) - * (AT", h ) be a m ap. F or each x G A f , we den ote T ^ M := Kerd f, and T j M : = its orthogonal com plem ent in (T ,M , g ). T h e elements o f

T ; M and T ^ M are called vertical resp. horizontal tangent vectors o f M at the

point * . Let us denote by ( ) v and ( ) u the orthogonal projections o f T ,M on

T ? M resp. T f M .

T h e m a p / is said to be horizontally conform al , if, V x G A / such that df, ^ 0, d f , : T * M - » Tf (,)N is a conform a l, linear isom orphism . For such maps we have (see Ref. |Ee-Le/83])

Chapter

J,

Seetioa 4. Urmpka o f hornetrie Immenioaa, Uoaformml

Map*, ...

29

A m ap / : (M , g) —* (TV, A) is said to be a harmonic m orp h itm , if, for a ny harmonic fu n ction 4> defined on an open set V o f TV, the com p osition <ft o f is harmonic on /-■ (V ) .

T h e following proposition , which we will use later on , is due to Fuglede [Fu/78] a n d Ishiara [Is/79] (see also Ref. [E e-L e/8 3 ]). ,

P r o p o s i t i o n 1 .4 A map f : (A /m, g) —» (T V , A) i$ a harmonic m orphitm t, iff

it it a harmonic and horizontally conform al map. I f f it non-conita nt, it it a

ttib m en ion on an open dense tubict o f M (and t o m > n ). I f at a point x

ra n k d f, < n, then dfa = 0.

L e t / : (M m,g ) -> (TV", A) be a submersion. T hen, T VM — M and T u M — M are sm ooth v ecto r bundles. Hence, in the neighbourh ood o f each p oin t o f M we m a y take an orthonorm al frame X i , . . . , X * ,X ,,+i , . , . , X m o f ( M ,g ) , such that

X t , . . . , X H e G ° ° (T UM ) and X m+u. . . , X m € G °°(T VM ) .

F o r all jr € / ( A / ) , the fibre Ft = f ~ l (y ) o f / at the p oin t y is a subm anifold o f M

o f dimension m — n w ith T»(F f ) = T j rM .

L e t the inclusion m ap tr : Ft —* (M , g) be an isom etric immersion. Its second fundamental form satisfies

V i . F( X . 1n ) = ( V x n ) “ , V X , r e c r >( T ''M ) , Vx e r , ,

w here V is the L evi-C ivita connection o f (A /, g). T h u s, the tension field o f tv is given by

V * > = £ 6 ! f M = ( r , ( F , ) | ‘ (1.24) ¿«■+1

a n d is equal t o m - n tim es the mean curvature o f the fibre Fr .

Sin ce / o i , is constant on Fr , V d ( / o if ) = 0, w e get [B arEe/81], using the com p osition law,

v < t r , ( x , r ) = - j / , ( V d . , ( x , y ) ) , v x , r e T , ( F , ) . ( ¡m)

T h e submersion / is said t o be Riem annian , if, V* € A f, dfM : T f M —*

Chapter 1, Sectioa 4. Graphs o f hornetrk Com formal Maps, . . . 30

P r o p o s i t i o n 1 .5 L et f : (M , g ) —» {N ,h ) he a tuhmertion. Then:

fa ) f hat totally geod etic fibre», iff ^4f\TyMxTvM = 0.

If, m oreover, f i t Riem annian, then alto (b) and (e ) hold:

(h) ^ df\THA/xT"M ~ 0*

(e ) the following con d ition t are equivalent: (i) f hat m inim al fibre t ; (ii) f it h a rm onic; (iii) f it a harm onic morphitm.

If, on the other hand, f i t a harmonic m orphitm , then (d ) and (e ) hold:

(d ) i f n = 2, the fib r et are minimal.

(e ) i f n > 3, the follow ing conditiont are equivalent:

Ii) the fibret are m inim al;

(ii) V « / it vertical everywhere;

(iii) the mean curvature o f the horitontal distribution, which it the vertical vector

field given by

Wx,X;)V,

is

equal to 2ef

Let now / : (AI m,g ) —* ( N " , h ) be a Riemannian submersion. From now on X » , . . . , X n , X m+ u • • •, X m denotes a local fram e o f ( M , g), such th a t X i , . . . , X n G

G ° ° (T UM ) and X „ + i , . . . , X m G C ° ° (T VM ) . N ote that from P rop. 1.5 w e have

T/ = f : V d f { X „ X l) = £ V 4 f ( X i ,X i ) . (1.26)

1-1 l-«+l

O n ( T u M ,g ) we have an induced conn ection which is given by:

V jf Z. = (VXZ,)“ , VZ e

x

e T,M .

P r o p o s i t i o n 1 .6 L et f : (M m, g ) - » (N n, h ) be a Riem annian tu b m ertion , and

denote by kTj the te ctio n o f T a M given by

* > - U r l r . J ' V / ) ■

Then we have:

(a )

v ,

e / ( A / ) , X e F „ # T ,(x ) = - r , , (x).

In particular,

||T/(

x

)||

a

=

||7jf

(*)||#- Thut, the fibret o f f have con ita n t mean

Chapter I, Sett toe 4. Graphs o f hornet ric h ₃₁

( t ) v » e f ( M ) , i f f , , X e T ,( F , ) = T ? M ,

( v > ' 0 " =

.

In particular, the fibre» o f f have parallel mean curvature, iff MTf it a parallel »cction o f T UM along the vertical vector field».

f < ) v x e o ° ° ( T “ M ) , < r (V x ( # r » ) = v ' - ' r , .

P roof. Let jr € / ( A / ) . From E q. (1 .2 4 ) we have

r , , M =

£ (V * ,X ,)f, V iE F ,,

i= n + l and from Eqs. (1 .26),(1.25)

*> (*) - E v ^ , ( x ltx . ) = - ^ ( V d ( . , ) , ( x , , x , ) )

= H W V .

Since T*f (x ) G T f f M , from the definition o f MTf we get —7Jf (x ) = MT/(x). Now let

X € T .( F ,) . Then,

V > ( r , f ) . = - V > ' ( , , >(*t» ) , = - V * ( * r , ) , .

Since (7«(F|)| m T f f M , Vx € F t , the connection ^ V 1 o f ( T ( / ,f ) ) A la exactly

equal to (V * 1 ) * . T hus, we have

X 1 (T ,,) . = ( v > " ‘ ( * , ) . ) ' = - V / y n » , ,

and we have proved (b ).

N ow we prove (c ). F or all X G G ° ° ( T M ) ,

= v i " ‘ ( ^ ( w » ) - V d f (x ,M T t ) = V x ‘ r , - V 4 H X , M r ,).

S o, from P rop . 1.5(b) we get

Chapter I, Section 4. Graph» o f hornet rh Immersions, Conformal M aps, . . . 32

Next we stud y the mean curvature H o f the graph 1 / o f / . N ote that the g,j (given in Eq. (1 .2)) are in this ca se given by

(1.27)

P r o p o s i t i o n 1 .7 L et f : (M m, g ) —► (N n,h ) be a Rtem anm an ttib m en ion . Then

we have:

(a ) m H = ( 0 , » » A .

In particular, f o r any point x G M , 7 )(x ) = 0 , iff H x = 0, and $o the follow ing condition» are equivalent:

(i) I ) it a m inimal submanifold o f M x N . (ii) f it harm onic.

(Hi) f it a harm onic morphism. (is) the fibres o f f are minimal.

(b ) The following conditions are equivalent: (i) Tj has constant mean curvature. (ii) ||7 /1|* is constant.

(Hi) the fibres o f f have constant mean curvature, the norm o f which is the same f o r all fibres.

( c ) ifT j has parallel mean curvature, then V * *7 / = 0, V X e C ° ° (T HM ) .

P ro o f. From L em m a 1.1 (ii),

T h u s, we have m H — ( 0 ,T /) A, and (a) is proved b y applying Lem m a 1.2(i) and P r o p . 1.5(c).

m H = - ( Z ,d f ( Z ) ) + (0 ,fV ) = (0 ,1V )1 ,

w here W and Z are given b y Eqs. (1.3) resp. (1 .4 ). From Eq. (1 .27 ),

lv = Ejv,< f(x„x1)+ f ;

v<v(x„x,j,

a n d from P rop . 1.5 and E q . (1.26) we have

w

= £

Vrj(x„x,) =

T , .

H ence,

Chapter

I,

See Horn 4. Crap ha d hornet rk Immenioaa, Coaformal Mapa, ...

33

The above expression fo r Z gives = £ " - i i < Tf i 4 f ( X i ) > * d f( X i) . Since dfa : T * h t —► T /(,)N is an isometry, dfs ( X i ) i . • • ,4 f* (X n) is an orthonorm al basis ° f 2/(,)JV . T h u s,

W )= \r, = \w ,

that is,

z = ( * U , r ’ <5

Therefore, m H — — ( Z , d f ( Z ) ) + (0 ,2 d f ( Z ) ) = ( - Z ,d f ( Z ) ) , and so

1 * * 1 0 * = l|Z||,> + ||^(Z)||i=2||Z||f*

= t w m i = | i r / i : .

C onsequently, 1 / has constant mean curvature, iff ||7/||* is constant, and (b ) follows from P rop . 1 .6 (a ).

Finally, we p r o v e (c). From Lem m a l .l ( i ii ) we have, V X G G ° ° (T M ),

= ( 0 , V j f V - V d f ( Z <X ) ) i .

If F/ has p a ra llel mean curvature, then, from Lem m a 1.2(i), = V d /( Z , X ) , that is,

V Tt, = \ i l ( l- X T „ X ) . Using P ro p . 1 .6 (b ) we obtain

V ' V , = 0, V i e 0° ° [T “ M ) . <9

Let / : ( A I m,g ) —* (iV " ,h ) be a harmonic morphism. W e are now going to study the m e a n curvature H o f T/.

Let U = ( i t € M : i f a5^ ®}* From P rop. 1.4, i f / is not con stan t, If is an open dense subset o f M , and, V* G U , dfa is a submersion with