Polynomial Harmonic Morphisms

Martin Svensson

Examensarb ete p oang

Lunds Universitet

Novemb er

Contents

Abstract

Acknowledgements

Chapter Intro duction

Motivation

History

Chapter Harmonic Maps

The Second Fundamental Form

Harmonic Maps

Harmonic Functions

Chapter Harmonic Morphisms

Horizontal Conformality

Harmonic Morphisms

The Existence Problem

Polar Sets

Chapter Polynomial Harmonic Morphisms

Globally Dened Harmonic Morphisms

The Classication of Ou

Polynomial Harmonic Morphisms of Higher Degree

Chapter Applications

The Theorems of Eells and Yiu

The Symb ol of Harmonic Morphisms

Bibliography 1

Abstract

The aim of this Masters thesis is to b e the rst survey of known

results on p olynomial harmonic morphims b etween Euclidean spaces

These were rst studied by Baird in in the early s He obtained

several results on the sub ject but left op en the still unsolved classica

tion of such maps In the article from Eells and Yiu classied

the homogeneous p olynomial harmonic morphisms whose restrictions

to spheres are again harmonic morphisms to spheres These are the

well known Hopf p olynomials of degree This result revitalized the

sub ject and so on thereafter Ou obtained a complete classication of

the homogeneous p olynomial harmonic morphisms of degree During

the preparation of this thesis a very interesting development has taken

place with Abab ou Baird and Brossard writing the article proving

that this is still a very active area of research

In Chapter we discuss the W eierstrass representation of minimal

surfaces as a motivating example and the history of general harmonic

some years ago morphisms b eginning with Jacobi

Chapters and are devoted to the intro duction of harmonic maps

and harmonic morphisms where we also derive some of their basic prop

erties

In Chapter we then study p olynomial harmonic morphisms We

show that every globally dened harmonic morphism b etween Eu

clidean spaces of suciently high dimensions is necessarily p olynomial

We give the complete classication due to Ou of those homogeneous

of degree and discuss some examples of higher degree A general

metho d for constructing nontrivial examples is provided and we make

a conjecture on the structure of p olynomial harmonic morphisms based

on known results on those of degree

In Chapter we use the results of Chapter to give a new pro of of

the ab ove mentioned result by Eells and Yiu regarding homogeneous

p olynomial harmonic morphisms b etween Euclidean spheres Finally

we show how the results derived so far can b e used to give information

concerning the singularities of general harmonic morphisms

has been my rm intention throughout this work to give references It

to the stated results and credit to the work of others The only results

I claim are mine wil l appear in chapter and and have been marked

Any statement example or proof left unmarked is with an asterix

considered to be too wel l known for a reference to be given 2

Acknowledgements

I am grateful to all those who help ed me improve this work by use

ful comments and suggestions in particular my sup ervisor Sigmundur

Gudmundsson to whom I am also grateful for his patience and encour

agement

Martin Svensson 3 4

CHAPTER

Intro duction

Motivation

The study of harmonic morphisms involves to a large extent the

study of harmonicity and minimal submanifolds two concepts which

themselves are strongly related In this section we illustrate this rela

tionship with an example from classical dieren tial geometry relating

the mean curvature of a surface in R to the Laplacian of the co ordinate

functions For details see

A regular parametrized surface is a C map

X U C R

where U is op en and connected The map X is assumed to have injec

tive dierential so that along the image X U in R we have a normal

vector eld

X X

u v

X X

j j

u v

It is customary to use the following notation

D E

2 2

X X X X

E F G

u u v v

and

D E D E D E

2 2 2

X X X

e N f N g N

2 2

u u v v

Then the mean curvature of the surface is dened as

eG g E f F

tracedN H

EG F

and X is said to b e a parametrized minimal surface if the mean curva

ture vanishes everywhere

b e a relatively compact subset of the domain U and h Now let V

b e a C function on V Then

X u v X u v thu v N u v

t 5

is called the normal variation of X V determined by h The following

very geometric result motivates the name minimal see do Carmo

for a pro of

3 2

Theorem Let X U R be a regular parametrized C

V surface Then X is minimal if and only if for every bounded V

U and every normal variation X of X V we have

AreaX V

t=0

2 3

It is well known that every regular C surface in R may lo cally

b e parametrized by isothermal co ordinates ie co ordinates for which

E G and F Let us therefore assume that X is isothermal Then

we have for the Laplacian X of X

D E D E D E

2 2

X X X X X

2 2

u u u u v

E D D E

2 2

X X X X

u u v u v

2 2

X X

u u u v

D E

and similarly X Hence X is normal to the surface

and

e g hN X i

E E

This implies the following

3 2

Theorem If X U R is a parametrized C surface with

X X X X

2 2

j j j j and h i then X is minimal if and only if X is

u v u v

harmonic

In the spirit of Theorem a minimal parametrized surface can b e

dened as a map

X U C R

satisfying

D E

2 2

X X X X

u v u v

and

X 6

If we in addition to this assume that U is simply connected it follows

from elementary complex analysis that there exists a holomorphic map

U C such that

X Re

alent to That X is isothermal is then equiv

1 3 2

2 2 2

z z z

Cho osing suitably leads us to the famous representation by Weier

strass

Theorem The Weierstrass Representation Let U be an

open simply connected subset of C and X U R a parametrized

surface satisfying

E D

2 2

X X X X

u v u v

and

Then there is exists pair of meromorphic functions f g in U such that

f and f g are holomorphic f g and

2 2

X z X z Re f w g w i g w g w dw

for al l z U Conversely every pair f g of meromorphic functions

as above dene a minimal parametrized surface in this way

In the next section we shall see how the results presented here

demonstrate a certain duality b etween minimal conformal immersions

and harmonic morphisms

History

The history of harmonic morphisms is generally thought to have

b egun with the article of Jacobi from on the solutions of

Laplaces equation in three dimensions Here Jacobi investigated nec

dened on an op en essary conditions for a complex valued function

subset of R such that for any holomorphic function f the comp osition

f is harmonic ie

A harmonic morphism though ought to b e a map that in some sense

preserves a harmonic structure It was for that purp ose more than 7

a century after Jacobi that harmonic morphisms were formally intro

duced by Constantinescu and Cornea in in the context of harmonic

spaces in abstract p otential theory

In general a harmonic space in the sense of Brelot see is a

lo cally compact Hausdor space X endowed with a sheaf H assigning

to each op en subset U of X a real subspace H U of the continuous

functions on U such that the following conditions are satised

X has an op en base for its top ology consisting of regular sets

A regular set is an op en relatively compact subset V of X with

nonempty b oundary V such that for every continuous function

f on V there is a unique element H H V which can b e

extended to V and equals f on V Furthermore if f then

If U X is op en and connected and fu g is an updirected

family in H U then either sup u is in H U or sup u

A A

For an op en subset U of X we call H U the harmonic functions on

U It is well known that R is a harmonic space with the harmonic

functions as solutions to Laplaces equation More generally every

Riemannian manifold is a harmonic space with the harmonic functions

as zeros to the LaplaceBeltrami op erator These results are essentially

e who showed see Chapter that the solutions due to R M Herv

to a uniformly elliptic equation

m m

X X

f f

b a cf

ik i

x x x

i k i

i=1

ik =1

with co ecients a b and c lo cally Lipschitz in a domain R

ik i

denes a system satisfying the axioms of a harmonic space

As dened by Constantinescu and Cornea in a harmonic mor

phism is a continuous map

X X

b etween harmonic spaces X and X such that for every op en U X

and harmonic function f on U the comp osition

U R f

is harmonic Since every harmonic function on C is lo cally the real part

of a holomorphic function we see that this is exactly what Jacobi was

investigating The aim of Constantinescu and Cornea was to generalize

results from the theory of Riemann surfaces to harmonic spaces with

harmonic morphisms replacing the holomorphic maps 8

Some decade after Constantinescu and Corneas article Fuglede

and Ishihara published indep endently their investigations on harmonic

morphisms in Riemannian geometry see and Their results

showed that in the sp ecial case when the harmonic spaces are Riemann

ian manifolds the harmonic morphisms are rich in geometric features

with several interesting applications and problems

If we return to Jacobi for a while assume that C is a

harmonic morphism where R is op en By cho osing f w w for

w C we see that is in fact smo oth Furthermore for a holomorphic

function f whenever the comp osition is dened we have

f f

2 2 2

w x y z w

Since f may b e chosen arbitrary we see in this case that the following

two conditions are necessary and sucient for to b e a harmonic

morphism

i The map is harmonic that is

2 2 2

x y z

ii the map satises

2 2 2

x y z

These conditions were obtained by Jacobi and b oth Fuglede and Ishi

hara noticed that they have natural generalizations to the case when

g N h is a map b etween arbitrary Riemannian manifolds M

These generalized conditions together remain necessary and sucient

for to b e a harmonic morphism

The condition i says that must b e a harmonic map Such maps

were intro duced as maps M g N h which are in the sense of

the calculus of variations the critical p oints of the energy functional

jdj E

The manifolds M and N are here assumed to b e compact and oriented

but the EulerLagrange equation of this variational problem makes it

p ossible to dene harmonic maps b etween arbitrary Riemannian man

ifolds The rst formal denition of a harmonic map was given by

Fuller in the year in after some preliminary work by Bo chner 9

and Morrey A thorough investigation of harmonic maps was also con

ducted by Eells and Sampson some decade later in their celebrated

article

If we write i in the sp ecial case of N C then the

1 2

second condition ii obtained by Jacobi is equivalent to

2 2

jgrad j jgrad j hgrad grad i

1 2 1 2

This means that for x either d or grad and grad

x 1 2

3 3

span a dimensional subspace of T R R which is mapp ed confor

mally onto T C C by d This is expressed by saying that is

(x)

horizontal ly weakly conformal

If we compare the equations i and ii with those obtained in the

previous section for minimal isothermal parametrized surfaces we see

the duality mentioned earlier the concept of a harmonic morphism is

in a sense dual to that of a harmonic conformal map

The harmonicity and the weak horizontal conformality give the the

ory of harmonic morphisms b oth analytic and geometric dimensions but

also make the question of existence very dicult since we are dealing

with an overdetermined nonlinear system of partial dierential equa

tions

So on after Fuglede and Ishihara several mathematicians followed

in the study of harmonic morphisms To solve the question of exis

tence attempts were made to classify harmonic morphisms in dierent

contexts Jacobi had himself investigated the conditions for a function

F F x y z w where x y z w R C such that every lo cal

solution w x y z to the equation

F x y z w

is a harmonic morphism He proved that this is true if F is holomorphic

in w and a harmonic morphism in the rst three variables In particular

he studied the case when the equation is given by

Aw x B w y C w z

where A B and C are holomorphic functions satisfying

2 2 2

A B C

In the late s Baird and Wo o d gave in a complete classication

3 2

of harmonic morphisms from domains of R to any Riemann surface N

This was one of the rst classication result for harmonic morphisms

and could b e seen as a complete solution to the problem p osed by

Jacobi Their result is essentially that every harmonic morphism 10

3 2

R N arises as a lo cal solution x y z of an equation

of the kind

2 2

hf w g w i g w g w x y z i

for two meromorphic functions f g N C fg With the duality

mentioned earlier in mind this should b e compared with Theorem

After this result was published the theory of harmonic morphisms

has grown rapidly as can b e seen on the Bibliography of Harmonic

Morphisms At present Baird and Wo o d are writing the rst

b o ok on the sub ject 11 12

CHAPTER

Harmonic Maps

Throughout this work we shall by a Riemannian manifold M g

mean a smo oth ie C real connected manifold of dimension m

together with a smo oth Riemannian metric g The dimension m is al

ways meant to b e nite We denote by r the LeviCivita connection

M k

of M asso ciated with the metric g and by its co ecients The

letters M N and P are reserved to mean Riemannian manifolds All

maps M N and functions M R are unless otherwise stated un

dersto o d to b e smo oth and so is any section of a smo oth vector bundle

V M over M We denote by C V the totality of such sections

In this chapter we present the basic notion of a harmonic map b e

tween Riemannian manifolds To do this in an invariant way we rst

intro duce the second fundamental form of a map in terms of vector

bundles and sections thus relating it to the common notion of the sec

ond fundamental form of an immersion and its mean curvature vector

For a deep er exp osition of harmonic maps we refer the reader to the

and rep orts and the b o oks

The Second Fundamental Form

In this section we shall dene the pullback bundle of a map b etween

Riemannian manifolds and equip it with a suitable Riemannian metric

together with a compatible connection

Denition Let M N b e a map The pul lback bund le

of is the bundle TN M over M with

TN fx v j x M v T N g

(x)

and

x v x for x M v T N

(x)

Thus TN is the induced vector bund le of TN by Obviously

if n is the dimension of N this is an ndimensional vector bundle over

M for x M cho ose a neighb ourho o d U of x in N and a smo oth 13

n 1

trivialization U R U where TN N is the canonical

pro jection Then

1 n 1 1

U R y v y y v U

1 1 1

is a smo oth trivialization of U A section V C TN

of the pullback bundle is by denition a map

V M TN

such that

V T N

(x)

for every x in M Thus for Z C TN x Z is an element

(x)

of C TN denoted by Z or simply Z Another imp ortant

example of a section of TN is the map

M x d X

x x

for a section X C TM of the tangent bundle of M

Denition By a smooth variation of M N we mean a

family of maps M N such that

t t 0

If is a smo oth variation of then

M x T N T N

(x) (x)

t=0

1 1

is a section of TN Conversely for a section V C TN

dene a family x exp tV If N is complete this will b e dened

t x

(x)

throughout M R and

dexp V V

(x) 0 x x

t=0

Thus we have

Prop osition If N is complete then for a map M N

every section in C TN is of the form

T N M x

(x)

t=0

for some smooth variation of

Since we have V T N for a section V C TN and

(x)

x M we may dene a Riemannian metric also denoted by h on

TN by

hV W x h V W

x x

(x)

for x M and sections V W C TN of the pullback bundle

Thus we have made TN into a smo oth Riemannian vector bundle 14

over M Our next step is to dene a connection on TN compatible

with h For X Y C TM and x M cho ose a curve

M with x and X Let P T M T M denote

x t x (t)

parallel transp ort along Then from the compatibility of the Levi

Civita connection on M we have

M 1

r Y x P Y

(t)

X t

t=0

It is therefore natural to make the following denition

Denition The pul lback connection of M N is the

connection

1 1

r C TM C TN C TN

on TN dened by

P V V x r

(t)

t=0

for x M X C TM V C TN and M a

curve with x X Here P T N T N is

x t

(x) ( (t))

the parallel transp ort along

It is a direct consequence of Denition and equation that

for Z C TN we have

N 1

Z r Z x P Z x r

(t)

d (X )

t=0

It is easy to see that r is a well dened connection on TN

and uniquely determined by equation Furthermore it is an easy

consequence of the fact that parallel transp ort is an isometry that r

is compatible with the metric h on TN see page and

page

Prop osition If M N is a map and X Y C TM

then

r dY r dX dX Y

X Y

Proof Since the left hand side is tensorial in X and Y it is enough

and Y for lo cal co ordinates to prove the statement for X

i j

x x

x around x M For that purp ose we cho ose lo cal co ordinates y 15

on N around y x Then

X X

d r r d r r

j i j i

i i j j

x x x y x y

x x

2 2

i j j i

x x x x y

r r

j i

i j

x y x y

The symmetry of the second derivatives implies that the rst sum in

the last expression vanishes The second is also zero since by equation

X X

r r

j j i

x y x x y

j i

x x y

x y

On the cotangent bundle T M of M we have a metric g obtained

M with their inverse images in T M under the by identifying T

isomorphism

T M Z g Z M T

x x

Thus if g X and g Y for some X Y T M then

x x x

g X Y g

On the tensor pro duct T M TN we may then dene a metric

h i by

h V W ix g h V W

(x)

for C T M and V W C TN Since the dierential

d of is a section of T M TN we get by denition

hd dix h d e d e

x i x i

(x)

trace hx

g 16

where e is any orthonormal base of the tangent space T M

i x

Recall that on T M we have a connection r dual to r given by

r Y X Y r Y

X X

for X Y C TM and C T M ie r is the ordinary co

variant dier ential of forms on M Thus we may dene a connection

on C T M TN by the following

Denition For a map M N r is the connection on

C T M TN given by

r V r V r V

for X C TM C T M and V C TN

That r is a well dened connection on T M TN is clear and

it is easy to verify that it will b e compatible with the metric h i

Denition For M N the second fundamental form of

is the covariant derivative rd of d by r

By denition we have

Y dY dr rdX Y r dY r

for X Y C TM Using Prop osition it is easy to see that the

second fundamental form

rd C TM C TM C TN

is symmetric and tensorial ie bilinear over the ring of smo oth func

tions M R

Denition For a map M N the tension eld of is

the trace of the second fundamental form of

trace rd

For maps M N and N P we write d rd for the

section d rd and rd d d for rd d d From the

chain rule we now deduce the following

Prop osition If M N and N P are maps between

Riemannian manifolds then

rd d rd rd d d

and

d trace rd d d 17

Harmonic Maps

We now have the prop er to ols for dening the concept of a har

monic map b etween Riemannian manifolds and to derive its funda

mental prop erties

Denition Let M and N b e Riemannian manifolds A map

M N is said to b e harmonic if its tension eld vanishes every

where

We also dene a stronger related concept

Denition Let M and N b e Riemannian manifolds A map

M N is said to b e total ly geodesic if its second fundamental form

vanishes everywhere

We see from Prop osition that the comp osition of two totally

geo desic maps is totally geo desic but that this need not b e true for two

harmonic maps As indicated in the previous section the theory has a

close connection with the calculus of variations We therefore pro ceed

to give a variational characterization of harmonic maps

Denition Let M and N b e Riemannian manifolds and as

sume that M is compact and oriented For a map M N the

energy functional is the integral

E jdj

where is the volume form of M and jdj hd di is the squared

norm of d as dened in the previous section The map is said to b e

a critical point of the energy functional if

t=0

for any smo oth variation of

Theorem Let M and N be compact Riemannian man

ifolds If M is oriented then a map M N is harmonic if and

only if it is a critical point of the energy functional

Proof We mainly follow Urakawa in Let b e a smo oth

variation of and write t x x M N Cho ose

a lo cal orthonormal frame e of the tangent bundle TM and write

i 18

e for e as vector elds of the pro duct manifold M By

i i

Prop osition we have

r de r d

For jtj dene X C TM by

g X Y hd dY

for an arbitrary Y C TM Then

X X

hde de hr de de

i i i i

i i

de hr d

e g X e hd e dr

i t i i

hd rde e

i i

divX hd de e

t i i

The integral of the rst term vanishes by Stokes theorem Thus we

obtain

h E

dt t

t=0 t=0

Since this holds for any smo oth variation of the statement follows

Remark Equation is generally refered to as the rst

variation As mentioned in the previous chapter harmonic maps were

originally intro duced as solutions to this variational problem Intu

ological irregu itively deforming in a manner that increases the top

larity Fuller page of will also increase the energy E

Thus the idea was to nd harmonic representatives in each homotopy

class of maps M N to b e used as homotopic normalizers of given

maps This was the main theme of the article of Eells and Sampson

where the following result was achieved

Theorem The EellsSampson Existence Theorem Let

M g and N h be compact oriented Riemannian manifolds where 19

N h has nonpositive sectional curvature Then any homotopy class

of continuous maps M g N h has an energy minimizing har

monic representative

For more details on the EellsSampson Existence Theorem and its

history we refer to the b o ok of Urakawa

For a function f M R and a lo cal orthonormal frame e for

the tangent bundle TM of M we have

f trace rdf

rdf e e

i i

f M

r df e df r e

i i

e e

i i

e e f r e f

i i i

grad f e g r

divgradf

and we recover the familiar LaplaceBeltrami op erator divgrad

on M For this reason we shall in the case of a function henceforth

write instead of

k m

Cho osing lo cal co ordinates x and y around p oints x M

and y N with y x for M N a straightforward calcula

tion gives

g rd

X X X

ij M k ij N ij

g g g

i j k i j

x x x x x

ijk ij

M ij

i j

x x

for n This implies that if N R then is harmonic if

and only if each of its comp onents are harmonic functions and

M 1 M n

Example Let I b e an op en interval of R I M b e

a regular curve on M and x lo cal co ordinates on M Then from 20

equation we get

2 k i j

d d d

k k

dt dt dt

Thus is harmonic if and only if it is a geo desic

m n

Example If M N is an isometric immersion we

may identify X C TM with dX C TN and consider

T M to b e a subspace of T N for x M Since for X Y T M

x (x) x

N M

r dY rdX Y dr Y

d(X ) X

and we see that rd is the second fundamental form of M in the

classical sense ie the orthogonal pro jection of r Y onto the normal

with the identication mentioned ab ove Recall that the space T M

mean curvature vector of M in N is the trace of the second fundamental

form divided by m and that is said to b e a minimal immersion of M

into N if the mean curvature vector vanishes Thus we have

Theorem An isometric immersion is minimal if and

only if it is harmonic

The name minimality is motivated by the fact that if M is compact

and orientable then is minimal if and only if is a critical p oint of

the volume functional

where is the volume form of M asso ciated with the induced metric

g h Actually for every smo oth variation by immersions of

one may prove see page that

h i V

dt t

t=0 t=0

There is a natural generalization of isometric immersions to weakly

conformal maps ie maps M g N h with h g for some

function

fg M R

called the conformal factor of The adjective weak indicates that

may take the value in which case d Theorem can now b e

generalized to the following

m n

Theorem Let m and M g N h be a con

formal immersion Then 21

a if m then is harmonic if and only if M is minimal in N

b if m then two of the fol lowing conditions implies the other

is harmonic

M is minimal in N

is a homothety ie its conformal factor is constant

Proof Denote by the conformal factor of so that h g

We dene a new Riemannian metric g on M as the pullback of h by

M M

g h If r and r are the LeviCivita connections of M g and

M g resp ectively then since g g one may easily deduce that

see page of

2 2 2

M M 2 2 2

r Y r Y X Y Y X g X Y grad

X X

for X Y C TM Here grad is the gradient in M g Thus if

fX X g is a lo cal orthonormal frame of M g we then have for

1 m

i m

2 2 M 2 M

X X X X r grad r

i i i i g

X X

i i

Since M g N h is an isometric immersion it follows that

M N N

X dX dr dX r r

i i i

X d(X ) d(X )

i i i

M N

X dX dr r

i i

X d(X )

i i

2 2 2

d grad X X

g i i

for i m Hence by summing over i we arrive at

mH m dgrad

which immediately proves the theorem

The concept of weak conformality is dual to that of horizontal weak

conformality which we will dene in the next chapter We will also de

rive a result corresp onding to Theorem for horizontally conformal

maps

Example Let M S b e a map into the unit sphere

n n1 n

in R i S R b e the inclusion map and i Then the

comp osition law gives

di trace rdid d

Note that the rst term on the right is tangent to the sphere and the

second is orthogonal to it Thus is harmonic if and only if is parallel 22

to ie if the tangential part of vanishes If f x jxj

x R then

f df trace rdf d d

hgrad f i trace rdf d d

h i jdj

Hence is harmonic if and only if jdj

n n1

Example If R n fg S is the radial pro jection

given by x x jxj we see using the notation of Example

that

n jxj x

2 2

Hence is a harmonic map and jd j n jxj

n n1

Example Let H n fg S b e given by x xjxj

where

n n

h i H f x R j jxj g

2 2

jxj

is the hyp erb olic space of constant curvature Then we get

2 2

jxj

x x n

jxj

2 2 2 2

Thus is a harmonic map with jd j n jxj jxj

m1 m

Example Let i S R b e the inclusion and let x

m1 m1

S Cho ose a lo cal orthonormal frame fe e g for TS

1 m1

around x and geo desics S k n such that

e x for all k Denoting by r x and the derivation

k k

r gives us in the direction normal to S and using that

at the p oint x

i m

m m1

If F R R is a function and f its restriction to S we get

m1 m1

S S

f F i

dF i tracerdF di di

F F

i F i m i

r r 23

In particular if F is a harmonic p olynomial homogeneous of degree p

then

f pp m f

Hence f is an eigenfunction of It is well known that all eigen

functions of arise in this way see page

m n

Supp ose that R R is a harmonic map with each of its

comp onent functions p olynomials homogeneous of degree p and that

m1 n1 m1 m

restricts to a map S S Denote by i S R and

n1 n

j S R the inclusions so that i j By equation

i pp m i

From Example we see that is harmonic and that

jdj pp m

2m M

Example Recall that for two Kahler manifolds M g J

2n N

and N h J a map M N is said to b e holomorphic or

M N

simply holomorphic if ie if dJ J d and holomorphic

M N

if ie if dJ J d If is either or holomorphic

holomorphic then is said to b e

M M M M

Since M and N are b oth Kahler r J J r and similar for

N Thus if M N is holomorphic and X Y C TM for

M N

simplicity writing J for b oth J and J

JY dJY dr rdX JY r

Y dY dr J r

J rdX Y

rdJ X Y

using the symmetry of rd In particular

rdJ X JX r dX X

Hence by cho osing a lo cal orthonormal frame fe e f f g

1 m 1 m

for TM with J e f for all i we get

i i

rde e r dJ e J e

i i i i

i=1

Thus we see that every holomorphic map b etween Kahler manifolds

is harmonic This was rst proved by Eells and Sampson in The

converse is not true take for instance C C dened by z z

1 2 24

z z Then is not holomorphic with resp ect to the standard Kahler

1 2

structures on C and C but harmonic since

2 2

z z z z

1 1 2 2

Example If G and G are Lie groups with biinvariant

1 2

Riemannian metrics g and g resp ectively and

1 2

G g G g

1 1 2 2

is a Lie group homomorphism then is a harmonic map in fact totally

geo desic To prove this denote by g and g the Lie algebras of G and

1 2 1

G resp ectively By identifying these spaces with the tangent spaces

at the neutral elements we get an induced Lie algebra homomorphism

d g g

1 2

satisfying dX x d X for any X g and x G Fur

x x 1 1

thermore it follows from the biinvariance of the metrics that for left

Y X Y and similarly on invariant vector elds X Y on G r

G Thus for X Y g we get

2 1

Y dY dr rdX Y r

X X

r dY dX Y

d(X )

dX dY dX Y

which proves the statement

The following very imp ortant result was proved by Sampson in

by applying the uniqueness theorem of Aronsza jn in to the lo cal

expression of the tension eld

Theorem The Unique Continuation Let M N

be a harmonic map If at some point of M al l the partial derivatives of

to any order vanish then is constant In particular if is constant

on an open subset of M then is constant on the whole of M

Harmonic Functions

In this section we state some well known results on harmonic func

tions needed later on Lo cally there exists a great variety of harmonic

functions on a Riemannian manifold as the following result by Ishihara

shows see Lemma It is based on an extension of a lemma

by Bers in and will play a crucial role in the next chapter 25

m k

Theorem Let x be a point in M x be normal coor

dinates on M centered at x and fc c g constants with c c

k ij ij j i

ijk =1

and c Then there exists a neighbourhood U of x in M and a

harmonic function f U R such that

f f

x c x c

k ij

k i j

x x x

for al l i j k m

Using Bers lemma Greene and Wu proved the next result

Theorem For any x M there is a chart U x

around x such that al l the coordinate functions x are harmonic

Denition A smo oth function f M R is said to b e

subharmonic if

The function f is said to b e superharmonic if f is subharmonic

The following result is a direct consequence of the ellipticity of

equation see Chapter

Theorem The Maximum Principle If f M R is a

subharmonic function having a maximum in an open subset of M then

f is constant

Corollary If M is compact then every subharmonic function

on M is constant

In the sp ecial case when M is orientable Corollary is due to E

Hopf and may very well b e proved without app ealing to the Maximum

Principle For since is a divergence Stokes theorem gives in this

case that

for any function f M R In particular if f is subharmonic then

this implies that f in fact is harmonic Furthermore an easy calculation

gives that

M 2 2 M 2

f jgradf j f f jgradf j

Hence if f is subharmonic we get

Z Z

M 2 2

f jgradf j

M M

so f is indeed constant 26

Note that together with Theorem the result of Corollary

implies that there are no compact minimal even immersed submani

folds of R

Example In harmonic function theory in Euclidean spaces

the following functions are very imp ortant

R n fg x log jxj

and

m 2m

R n fg x jxj

where m It is easy to see that they are b oth harmonic where

dened If R is op en and connected and a then by a theorem

of Bo chner see any function u which is harmonic in n fag and

p ositive near a is of the form

x v x c log jx aj if m

and

x v x cjx aj if m

for some constant c and a harmonic function v in In particular

if u is harmonic and nonnegative in R n fg with n it follows

from an application of the Maximum Principle that v is in fact constant

Hence

ux b cjxj

for some constants b and c Since log jxj is not b ounded for large jxj

the same argument can not b e applied to a function harmonic and non

negative in R n fg Indeed any such function f must b e constant

by the theorem of Liouville see since the function f e z C

would b e harmonic and nonnegative in R 27 28

CHAPTER

Harmonic Morphisms

In this chapter we dene harmonic morphisms and prove some basic

facts on these The key to the theory is Theorem which states

that harmonic morphisms constitute a certain sub class of the harmonic

maps having the additional prop erty of b eing horizontally weakly

conformal We therefore b egin with a description of this concept In

the last two sections we briey discuss the existence and nonexistence

of harmonic morphisms and study their b ehaviour on p olar sets All the

results given in this chapter can b e found in a regularly up dated

list of publications on harmonic morphisms

tal Conformality Horizon

A map b etween Riemannian manifolds of equal dimension is con

formal if its dierential at any p oint is a conformal linear map with

resp ect to the Riemannian metrics Horizontal conformality is a gen

eralization of this concept to the case when the target manifold is of

lower dimension than the domain

Denition Supp ose that M N is a map At each p oint

of is the kernel ker d of the dierential x M the vertical space V

x x

d of at x The horizontal space H is the orthogonal complement

x x

of V in T M with resp ect to the Riemannian metric g on M

x x

If the map is a submersion ie d is surjective at each p oint of

M then we may asso ciate to two distributions on M the vertical

and the horizontal distribution asigning to each p oint x M the sub

spaces V and H of T M resp ectively A vector eld X on M is then

x x x

said to b e vertical horizontal if it b elongs to the vertical horizontal

distribution

By the Inverse Function Theorem the bre x is a p ossibly

disconnected submanifold of M for every x M the tangent plane

of which is the vertical space Thus the vertical distribution V is inte

grable and the map determines a foliation of M which leaves are the

bres of From the integrability of V one may easily deduce that the

horizontal distribution H is in fact smo oth on M

Two vector elds X C TM and Y C TN are said to 29

b e related if d X Y for every x M A vector eld

x x

(x)

X C TM is called basic if it is related to some vector eld

Y C TN If in addition X is horizontal then X is called a hori

zontal lift of Y It follows easily that for a given Y C TN there

exists a unique horizontal lift of Y which in fact is smo oth

m n

Denition Let M N b e a map The critical set

of is the set C fx M j d g The map M N is

said to b e horizontal ly weakly conformal if for each x M n C the

restriction of d to the horizontal space H is surjective and conformal

x x

8 4

Example Let R R b e the multiplication of quater

nions given by

x x x x x x x x x x x

1 2 8 1 5 2 6 3 7 4 8

x x x x x x x x

1 6 2 5 3 8 4 7

x x x x x x x x

1 7 2 8 3 5 4 6

x x x x x x x x

1 8 2 7 3 6 4 5

of quaternions then H H H If we identify R with the space H

q q q for q q H In the canonical bases we is given by q

1 2 1 2 1 2

have

x x x x x x x x

5 6 7 8 1 2 3 4

B C

x x x x x x x x

6 5 8 7 2 1 4 3

B C

(q q )

1 2

x x x x x x x x

7 8 5 6 3 4 1 2

x x x x x x x x

8 7 6 5 4 3 2 1

Clearly the horizontal space is generated by the rows of this matrix

2 2

Note that these are orthogonal and of equal length jq j jq j so if

1 2

q q and we express a horizontal vector v in the base given

1 2

by these rows as v then

1 2 3 4

2 2

d v jq j jq j

(q q ) 1 2 3 4 1 2

1 2

Thus for horizontal vectors v w we see that

2 2

hd v d w i jq j jq j hv w i

1 2

(q q ) (q q )

1 2 1 2

Hence is horizontally weakly conformal with the critical set con

sisting of the origin

The adjective weak indicates that the critical set of may b e non

empty From now on we shall refer to a horizontally weakly conformal

map as a horizontally conformal map it b eing understo o d that it need

not b e a submersion 30

The horizontal conformality of implies that there exists a function

M n C R such that for all x M n C and X Y H

xg X Y hd X d Y

x x

For x M n C and an orthonormal basis e of T M we have

i x

2 2

n x hd e d e jd j

x i x i x

i=1

where n is the dimension of N Hence extends in a unique way to

a continuous function on the whole of M also denoted by satisfying

equation This extended function vanishes on C and is called

the dilation of By equation is smo oth on the whole of M

Example If n N then the Lie group S C acts freely on

2n+1 n+1

the unit sphere S C by left multiplication and the resulting

2n+1 1

quotient space S S may b e identied with the complex pro jective

n n+1 2n+1

space C P the set of complex lines in C More precisely S is a

n 1 2n+1

principal bre bund le over C P with structure group S If S

n n

C P is the canonical pro jection then is a submersion so C P may

b e given a Riemannian metric induced by With this metric is

a Riemannian submersion ie horizontally conformal with constant

dilation The bres are of course the orbits under the action of

1 i

x fe x j R g

2n+1

for x S The tangent space of the bre at x is the vertical space

of and so is the real line through x spanned by ix and the horizontal

2n+1

space is the orthogonal complement of this in T S

Prop osition Let V W be a nonzero linear map

between nitedimensional Euclidean vector spaces and W V

be its adjoint characterized by h w v i hw v i Then is hori

zontal ly conformal if and only if is conformal If in addition dim

W then is a conformal isomorphism V dim

Proof Supp ose is horizontally conformal with dilation

Cho ose orthonormal bases fv v g and fw w g of V and W

1 m 1 n

resp ectively such that v w for i n and v if i

i i i

for i j n Hence n Obviously h w v i hw v i

ij i j i j

w v for i n so it is clear that is conformal The

i i

converse follows with the same choice of bases and the last statement

is obvious 31

m n

Example If M g N h is a map x M cho ose

lo cal co ordinates x and y on M and N around x and x y

resp ectively It is then easy to see that

d h y dy y grad x

Dene a lo cal frame on N by

e h n

Then he e h and d e grad By Prop osition

is horizontally conformal at x with dilation x if and only if

2 ij

x h g

i j

x x

for n It follows that is horizontally conformal with

dilation if and only if this holds for any x M and lo cal co ordinates

around x and x resp ectively In particular if y are normal at

x then is horizontally conformal at x with dilation x if and

only if grad n are mutually orthogonal and of equal

length x When N is a Riemann surface with a Hermitian metric

1 2

then by writing i in a lo cal holomorphic co ordinate we see

that the horizontal conformality is given by

g grad grad

In lo cal co ordinates x on M this can b e expressed by

i j

x x

In the interesting case when M is also a Riemann surface with a Her

mitian metric and z x iy a complex co ordinate on M then equation

reduces to

2 2

x y z z

In particular if is holomorphic then is automatically horizontally

conformal 32

Harmonic Morphisms

We can now dene harmonic morphisms and prove some of their

elementary prop erties

Denition A map M N is said to b e a harmonic

morphism if for every op en subset U of N with U and every

harmonic function f U R the comp osition f U R is

harmonic

The ab ove denition means that harmonic morphisms pull back

germs of harmonic functions to germs of harmonic functions It is

therefore immediate that the comp osition of harmonic morphisms is

a harmonic morphism A useful characterization for harmonic mor

phisms is provided by the next theorem which gives the link b etween

horizontal conformality and harmonic morphisms

Theorem The FugledeIshihara Characterization

A map M N is a harmonic morphism if and only if it is harmonic

and horizontal ly conformal

m n

Proof Supp ose M N is a harmonic morphism If

x M consider systems of lo cal co ordinates x and y around x

0 0

and y x resp ectively where we assume that y are normal

0 0

centered at y According to Theorem we may for every

n cho ose a function f dened and harmonic in a neighb orho o d

of y with

f f

y and y

0 0

y y y

for all n By assumption the function f is harmonic

in a neighb ourho o d of x so by Prop osition

f df rdf d d

In particular since at x

rdf dy dy and df

y y

the ab ove relation implies that x Since this holds for any

n and x M we conclude that the map is harmonic

To prove the horizontal conformality of we once more apply

Theorem by noting that we may for every sequence c

with c c and c cho ose a harmonic function f such

that

f f

y and y c

0 0

y y y

for all n Hence we obtain at x

tracerdf d d

c g

i j

x x

X X

ij ij

c g c g

i j i j

x x x x

= ij

where we as usual write for y We subtract

1 1

c g

i j

x x

from equation and obtain

1 1

X X

ij ij

c g c g

i j i j i j

x x x x x x

It now follows that

g for all and

i j

x x

1 1

X X

ij ij

g g for all

i j i j

x x x x

ij ij

This can b e summarized as

ij 2

g x for all n

i j

x x

The last system of equations is equivalent to the statement that the

comp onents all have orthogonal gradients of equal length so by

Example the map is horizontally conformal

Conversely if is harmonic and horizontally conformal with dila

tion U an op en subset of N with U and f U R a

harmonic function then we have f trace rdf d d on

1 1 i

U If x U consider once again systems x of lo cal and

y of normal co ordinates around x and x resp ectively

0 0 34

Then at x

tracerdf d d g

i j

y y x x

g grad grad

y y

2 M

x f

Thus is a harmonic morphism

m n

Corollary If M N is a nonconstant harmonic

morphism then is a submersion on an open dense subset of M so

m n

Proof We see that for x M if rank d n then d

x x

Hence the set of x M with rank d n is op en and nonempty It

is also dense for if on an op en subset we had d then by Theorem

we have d in the whole of M so would b e constant on

Corollary A harmonic morphism preserves harmonic

maps ie if M N is a harmonic morphism N P a

harmonic map then M P is harmonic

dim M At a p oint x M cho ose Proof Let n dim N

orthonormal bases e for T M and f for T N such that d e

i x j (x) x i

xf i n and d e for i n This is p ossible since is

i x i

horizontally conformal Thus at x we get for any map N P

tracerd d d

rd de de

i i

i=1

which immediately proves the statement

A submersion is always an op en map as follows from the Implicit

Function Theorem In view of Corollary the following result is

therefore not surprising For two dierent pro ofs see and 35

Theorem Every nonconstant harmonic morphism is

an open mapping

Corollary If M N is a nonconstant harmonic

morphism and M is compact then so is N and M N

Proof By Theorem the image M is b oth op en and closed

in N The result now follows from the connectivity of N

Example The result of Theorem gives us simple means

to determine whether a given map is a harmonic morphism or not

From Example we see that U R C U op en is a harmonic

morphism if and only if it satises the following system of dierential

equations

R 2

For a map M C a necessary and sucient condition for to b e

a harmonic morphism is that this holds for every lo cal co ordinate x

M R

replaced with and the second equation replaced on M with

with

From Example we know that any holomorphic map from

a Kahler manifold to a Riemann surface with a Hermitian metric is

harmonic Since the dierential of such a map will either commute or

anticommute with the almost complex structure it is not hard to see

that it will b e horizontally conformal see Corollary Hence

any holomorphic map from a Kahler manifold to a Riemann surface

with a Hermitian metric is a harmonic morphism Example shows

that the converse is not true in general

Example A Riemann surface N can b e dened as an ori

entable surface with a conformal class of Riemannian metrics A metric

b elonging to that class is then said to b e compatible with N

If M g N h is a harmonic morphism to a surface it

follows from Theorem that if h is a Riemannian metric on N which

is conformally equivalent to h then M g N h is a harmonic

morphism In particular the concept of a harmonic morphism from

M g to a Riemann surface is well dened

Example Let M M and N b e Riemannian manifolds and

1 2

assume that

M M N

1 2

is a map where M M is given the pro duct structure and pro duct

1 2

metric Dene for x M and y M the map M N by

1 2 x 2 36

y x y and similarly M N From the denition of the

x y 1

pro duct metric it follows that the inclusions i of M fy g and i of

1 1 2

fxg M into M M resp ectively are totally geo desic emb eddings

2 1 2

Using Prop osition gives

x y tracerdx y

tracerddi di x tracerddi di y

1 1 2 2

x y

y x

It follows that if and are harmonic for every x M and y M

x y 1 2

then is harmonic Furthermore for a function f M M R

1 2

gradf x y gradf x gradf y

y x

and the two terms on the right are orthogonal Thus we see that if for

every x M and y M the maps and are b oth horizontally

1 2 x y

conformal with dilations and resp ectively then is horizontally

x y

conformal with dilation satisfying

2 2 2

x y x y

y x

In particular if is a harmonic morphism in each variable separately

then is a harmonic morphism

Denition A map M N is said to b e horizontal ly

homothetic if is horizontally conformal and the gradient grad of

the square of the dilation is vertical everywhere

The horizontal homothety implies that the dilation is constant along

horizontal curves The concept of horizonal homothety is actually more

natural than it may seem at rst and it is in many ways a more suitable

generalization of a Riemannian submersion than that of a horizontally

conformal map For a further investigation on the geometry of hori

zontally homothetic maps see Chapter of By a result of Fuglede

we have the following

Theorem A nonconstant horizontal ly homothetic har

monic morphism is a submersion

The following result gives the theory nice geometric features and is

dual to that of Theorem It was rst proved by Baird and Eells

but the pro of presented here was given by Gudmundsson in the context

of semiRiemannian manifolds

m n

Theorem Let M N be a horizontal ly conformal

submersion Then

a if n its bres are minimal if and only if it is harmonic

b if n two of the fol lowing conditions imply the other 37

is harmonic

has minimal bres

is horizontal ly homothetic

Proof Let fZ Z g b e a lo cal orthonormal frame for

1 n

TN and for i n let X b e the horizontal lift of Z If is the

i i

dilation of then fX X g is a lo cal orthonormal frame for the

1 n

horizontal distribution H If X Y are basic vector elds on M which

are related to X and Y resp ectively and if H denotes the pro jection

on the horizontal space then we get

M M

H r Y g r Y X X

k k

X X

2 M

g r Y X X

k k

X g Y X Y g X X X g X Y

k k k

g X Y X g X X Y g Y X X X

k k k k

2 2 2

hY Z X X hY Z hZ X Y

k k k

2 2 2

Y hZ X hX Y X Z hX Y

k k k

hX Y Z hZ X Y hY Z X X

k k k k

Hence

N M

Y r dr Y

2 2 2

X Y Y X hX Y X Z

k k

In particular for X Y X we obtain

M 2 2

2 N

dgrad dr X X Z r Z

H k k k k

X Z

k k

2 N

dgrad r dX X Z

H k k k

This gives us

rdX X

H k k

k =1

rdX X

k k

k =1

2 N M

r Z dr X

k k

Z X

k k

k =1

2 2

dgrad dX X

H k k

k =1

ndgrad

If fV V g is a lo cal orthonormal frame for the vertical distri

n+1 m

bution we nally come to the conclusion

rdV V

H k k

k =n+1

V dr

k H

k =n+1

V d r

k H

k =n+1

ndgrad m ndH

where H is the mean curvature vector eld of the bres The theorem

now immediately follows from the last equation

As a direct consequence of Theorem we obtain the following

classical result of Eells and Sampson

Corollary A Riemannian submersion has minimal

bres if and only if it is harmonic

Example A simple example of a harmonic morphism is the

orthogonal pro jection followed by a multiplication of a scalar

m n

R x x ax x R

1 m 1 n

for some a R n fg It has trivially constant dilation jaj totally

geo desic bres and integrable horizontal distribution 39

Example Both the maps and from Examples and

are harmonic morphisms since they are horizontally conformal

1 2

with dilations x jxj and x jxj jxj Both have totally

geo desic bres and integrable horizontal distributions

Example Let G b e a Lie group with a biinvariant metric

and K b e a Lie subgroup of G Denote by G GK the canonical

pro jection onto the homogeneous space GK We may in a canonical

way turn GK into a Riemannian manifold with a Ginvariant metric

such that is a Riemannian submersion For leftinvariant vector elds

X Y on G it is well known that

r Y X Y

In particular if X and Y are leftinvariant vector elds on K then the

horizontal comp onent of r Y is zero Thus K is a totally geo desic

submanifold of G and it follows that has totally geo desic bres By

Theorem the canonical pro jection G GK is a harmonic

morphism

m m

Example If M N is a conformal submersion b e

tween Riemannian manifolds of equal dimensions m then we get from

the pro of of Theorem

mdgrad

2 2

It follows that a map M N is a harmonic morphism if and

only if is conformal If M and N are Riemann surfaces we know from

Example that every holomorphic map is a harmonic morphism

Conversely if is a harmonic morphism b etween Riemann surfaces

we see from equation that for every x M either z x

or zx Now and hence also z and z are real an

alytic by the ellipticity of equation so will b e holomorphic on

M Thus a map b etween connected Riemann surfaces is a harmonic

morphism if and only if it is holomorphic

We also deduce the following

Theorem If M N is a map between Riemannian

dimensions m then is a harmonic morphism manifolds of equal

if and only if is a homothety ie conformal with constant dilation

Proof If is a harmonic morphism with dilation which is not

constantly zero on M set M M n C which is an op en dense subset

of M The restriction of to M is a harmonic morphism and from

equation it follows that grad on M and then by continuity 40

on the whole of M Hence is constant Conversely if is conformal

with constant dilation then is harmonic by equation

m 0 m+1

Example Represent S n S in R as

m 0 m1

S n S fcos t sin t e j t e S g

m 0 m1

Dene S n S S as the pro jection onto the equatorial

hyp ersphere along the longitudes

cos t sin t e e

The horizontal curves for are those for which t is constant so is

horizontally conformal with dilation

cos t sin t e sin t

Hence is horizontally homothetic and since it has minimal bres it

follows from Theorem that is a harmonic morphism It has inte

grable horizontal distribution with small spheres sin tS as integral

submanifolds It is clear that cannot b e extended continuously to the

whole of S

b e any of the normed division algebras R Example Let D

C H or Cay of real dimension d or resp ectively The Hopf

2d d +1

polynomials are then dened as the maps R R given by

2 2

z z jz j jz j z z

1 2 1 2 1 2

for z z D These are all harmonic morphisms dened by ho

1 2

mogeneous p olynomials of degree with dilation x jxj Their

bres are spheres of dimension d so they cannot b e minimal in R

for d This shows that there is no equivalence in the last part of

Theorem

We see that the Hopf p olynomials all restrict to maps

2d1 d

S S

called the Hopf brations They are horizontally conformal with con

stant dilation see the pro of of Theorem and they are harmonic by

Example or by Theorem since their bres are totally geo desic

2d1

in S Hence they are submersive harmonic morphisms surjective

by Corollary

There are several ways of describing the Hopf brations see

and they have b een of great imp ortance for the development of homo

topy theory and bre spaces As we shall see in the next chapter the

Hopf brations and the Hopf p olynomials are also of great interest in

the study of harmonic morphisms

3 2

If we have a lo ok at the case of d denoting by S S 41

the corresp onding Hopf bration we make the standard identication

2 1 3 1 3 1

of S with C P Considered as a map S C P S S is easily

seen to b e the quotient map taking z z to the complex line in C

1 2

through this p oint This is the way the Hopf brations generally are

presented

If we identify S with SU via the map

z z

1 2

S z z SU

1 2

z z

2 1

we note that the horizontal space of at the identity of SU is

spanned by

and Y X

and the vertical space by

Since

X Y XY Y X V

the horizontal distribution of is not integrable This is actually true

for all but the rst of the Hopf brations as will b e motivated in the

following section

The Existence Problem

There is no general existence theory for harmonic morphisms in

deed by Theorem harmonic morphisms are solutions to an over

determined nonlinear system of partial dierential equations thus

making the question of existence rather dicult In most cases the

only known way of proving existence of nonconstant harmonic mor

phisms b etween given Riemannian manifolds is by a direct construction

of examples Another way is to give harmonic morphisms as implic

itly dened solutions to certain nonsingular equations a metho d that

go es back to Jacobi see Chapter This was used by Gudmunds

son in to construct complex valued harmonic morphisms dened

lo cally in several irreducible Riemannian symmetric spaces In some

noncompact cases even globally dened solutions were found Similar

metho ds have b een used by Baird and Wo o d in and to construct

b oth globally and lo cally dened complex valued harmonic morphisms

from Euclidean spaces We shall see in the next chapter that interest

ing existence theorems for p olynomial harmonic morphisms have b een

achieved In contrast to nd globally dened harmonic morphisms 42

b etween compact Riemannian manifolds has oered great diculties

Only a few examples are known among them the Hopf brations of

Example see The Atlas of Harmonic Morphisms When the

codomain is a compact Riemann surface though we have the following

interesting existence result by Gudmundsson

Theorem Let N be a compact Riemann surface and

n 2

a homotopy class of S bund les over N Then there exists a bund le

M N in and a Riemannian metric g on M such that the

bund le map M g N is a harmonic morphism

More is known ab out the nonexistence of harmonic morphisms

For instance the existence of a nonconstant harmonic morphism

m n

M N immediately gives necessary conditions on the dimensions

m and n namely that m n In the next chapter we shall derive some

further necessary conditions on the dimensions for the existence of so

lutions to the problem It is also clear that Corollary constitutes

a simple but imp ortant nonexistence result

Wo o d observed in that if M N is a horizontally con

formal submersion with integrable horizontal distribution H then the

asso ciated foliation F is total ly umbilic in M This means that for

X do es not a horizontal unit vector eld X the vertical eld V r

dep end on the choice of X For horizontal vector elds X and Y we

have in this case Theorem of

Y V r g X Y V grad

If in addition m n and is a harmonic morphism with totally

geo desic bres then F is spherical ie for every leaf L F the

H H

mean curvature vector H of L in M is parallel in the normal bundle

of the leaf Gudmundsson generalized ONeills fundamental equations

for a submersion see to the horizontally conformal case Using

this and a characterization of the totally umbilic spherical foliations on

manifolds with constant sectional curvature he proved in Corol

lary the following nonexistence result

m n m n

Theorem If m n and M N S S

m n m n

R S or H S then there are no nonconstant harmonic mor

phisms from M to N with total ly geodesic bres and integrable hori

zontal distribution

The last result implies that none but the rst of the classical Hopf

brations of Example have integrable horizontal distribution

A natural way to imp ose necessary curvature conditions on the

manifolds for the existence of nonconstant harmonic morphisms is by a 43

Bochner technique This metho d can briey b e describ ed as developing

an equation relating the Laplacian of bundle valued sections to the

curvature of the bundle in question known as a Weitzenbock formula

see From this formula vanishing results involving the curvature

can b e derived This was done for harmonic maps by Eells and Sampson

in and for harmonic morphisms by Mustafa in Using this

Mustafa then proved the following imp ortant theorem

Theorem Let M be a compact Riemannian manifold

with nonnegative Ricci curvature and let N be a compact Riemann

surface of genus Then any harmonic morphism from M to

N is constant

By cho osing suitable Riemannian metrics on the space M and ar

guing as in Example Mustafa obtained the following

There are no nonconstant harmonic morphisms from a compact

irreducible Riemannian symmetric space to a compact Riemann

surface of genus

There are no nonconstant harmonic morphisms from a compact

connected Lie group with a biinvariant metric to a compact Rie

mann surface of genus

In Mustafa develop ed the Bo chner technique further to non

compact domains in the language of moving frames and derived the

following astonishing result

Theorem There is no nonconstant global ly dened har

monic morphism from R into a Riemannian manifold with scalar cur

vature bounded above by a negative number

As a direct consequence there are no nonconstant harmonic mor

m n

phisms from R into the hyp erb olic space H

Polar Sets

In this section we shall return to p otential theory for some elemen

tary denitions needed in the next chapter Note that the functions

and maps are here unless otherwise stated not assumed to b e smo oth

Recall that if X is a harmonic space see Chapter and H its sheaf

of harmonic functions then for every regular set V in X and continu

ous function f V R there is a unique harmonic function H on

V which can b e extended to V so that the extension equals f on the

b oundary V Furthermore if f then H From this it easily

follows that for every x V the map

f H x

f 44

is a p ositive linear functional on the set of continuous functions on V

By the Riesz Representation Theorem there exists a unique Radon

measure on V so that

V V

H x f y d y

f x

for every continuous function f on V The measure is called the

harmonic measure for V at the point x

Example Let denote the normalized volume measure on

m1 m

the unit sphere S If B is the unit ball in R and f a continuous

function on the b oundary B S then it is well known that the

Poisson integral see

jxj

f y d y B x

jx y j

is harmonic and may b e extended continuously to B so that the ex

tension equals f on S Thus the harmonic measure for B at x

jxj

y d y d

jx y j

We now extend Denition to a much more general situation

Denition Given an op en set U in a harmonic space X a

function f U is said to b e superharmonic if

i f is lower semicontinuous

ii f is not identically on any comp onent of U and

iii for every regular set V V U and every x V we have

y f x f y d

It was shown by Herve see Chapter of that this denition

is consistent with the one given in Chapter The following concept is

classical in p otential theory

Denition Let X b e a harmonic space A subset E X is

said to b e polar if for every p oint x X there exists a neighb ourho o d

E U and a sup erharmonic function f in U such that f on U

Although p olar sets are generally small enough to b e ignored we

will need some basic facts on these in the next chapter

Example If n then any p oint of R is p olar More

generally if n every ane subspace A R of co dimension at least

is p olar Furthemore the p oints of any relatively compact subset U 45

of R are p olar in U These results can b e found in any elementary

b o ok on p otential theory see eg

Example For a nonconstant harmonic morphism b etween

Riemannian manifolds we know that its critical set C is closed and

nowhere dense It was proved by Fuglede in that it is actually

p olar

If M N is a harmonic morphism b etween Riemannian man

ifolds and f is a smo oth sup erharmonic function on N then we have

by equation

M 2 N

f f

Hence f is sup erharmonic Furthermore Constantinescu and Cornea

proved that a general harmonic morphism b etween harmonic spaces

pulls back sup erharmonic functions to sup erharmonic functions see

Corollary This implies the following result needed for one of

our main results of the next chapter

Prop osition The pul lback of a polar set by a non

constant harmonic morphism is again a polar set 46

CHAPTER

Polynomial Harmonic Morphisms

In this chapter we study p olynomial harmonic morphisms b etween

m n

Euclidean spaces By a polynomial map R R we mean a map

with p olynomial comp onents Its degree is the maximal degree of these

comp onents In Section we study general globally dened harmonic

morphisms b etween Euclidean spaces Why this is done in the con

text of p olynomial harmonic morphisms is motivated by Theorem

which states that if n every globally dened harmonic morphism

m n

R R is p olynomial Section is devoted to a classication of

the homogeneous p olynomial harmonic morphisms of degree due to

Ou and nally in Section we discuss some homogeneous p olynomial

harmonic morphisms of higher degree

Globally Dened Harmonic Morphisms

Globally dened harmonic morphisms b etween Euclidean spaces

have some very characteristic features at least if the dimensions are

large enough We shall here discuss some striking results on such maps

all of which were presented by Abab ou Baird and Brossard in the very

recent manuscript

m n

Theorem If n and R R is a nonconstant

harmonic morphism then is surjective

n 2n n

Proof For any a R the function x jx aj x R n fag

is harmonic If there was some a R not in the image of then

x jx aj

would b e a p ositive harmonic function on the whole of R and there

fore constant by the theorem of Liouville see Hence would b e

constant which leads to a contradiction

Example The map

C z e C

is obviously not surjective but a harmonic morphism since it is holo

morphic This simple example shows that the statement of Theorem

is not true for n 47

Next we show a Liouville typ e of theorem for harmonic morphisms

Theorem Every global ly dened harmonic morphism

m n

R R with n is a polynomial map of degree

To prove Theorem we need the following p otential theoretic

lemma For a pro of of this using probabilistic arguments see Lemma

Lemma Suppose that m and P is a closed polar set

m m

in R If f R n P R is a positive harmonic function then there

is a constant c such that

m2 1

f x c jxj

for al l x R n P

Proof of theorem If m n then by Corollary the

map is constant and we are done Assume m n If h R n fg R

is the harmonic function hx jxj then the comp osition

m 1

h R n R

is a p ositive harmonic function and by Prop ositon dened o a

closed p olar set By Lemma there is a constant c such that

n2 1 m2

jxj c jxj

m 1 n

which obviously will hold for all x R If we write

then this implies that for any i n we will have

i m2

n2 n2

j xj a jxj a jxj

m i

for all x R and some p ositive constant a Each is harmonic so by

the Cauchy estimates see and the ab ove inequality it follows that

the p ower series expansion of around is a p olynomial of degree

The following example shows that the condition n on the di

mension of the target manifold in Theorem can not b e removed

Let C C b e any holomorphic function and dene

F z cos z

for z C Since F is holomorphic it is a harmonic morphism and not

p olynomial except in trivial cases

For obvious reasons the next result is together with Theorem

very imp ortant in the study of globally dened harmonic morphisms

b etween Euclidean spaces 48

Theorem Every horizontal ly conformal polynomial F

m n

R R with n is harmonic

Here we give our corrected version of the pro of contained in the

original manuscript of

Proof It is enough to prove the statement for n For if

m n

the result holds in this case and F F F R R is

1 n

a horizontally conformal p olynomial where n then F F is a

i j

m 2

horizontally conformal p olynomial R R for any indices i j

Hence F is harmonic for every i so F is harmonic

Assume that F is a complex valued horizontally conformal p olyno

mial on R If F is constant we are done Otherwise denote the part

of F that is homogeneous of degree i by Q hence

F Q

i=0

where d is the degree of F By cho osing suitable co ordinates x in R

we may assume that is not a singular p oint of F and that

Q x x x ix

1 1 m 1 2

We write z x ix From the horizontal conformality of F we get

1 2

i=1

F F F

z z x

X X X

Q Q Q Q

r s r s

z z x x

i i

rs rs

X X

Q Q Q Q

s s r r

hr Q r Q i

x r x s

z z z z

rs rs

where we have written

Q Q

r r

r Q

x r

x x

3 m

Changing the order of summation implies that

p p1

X X X

Q Q Q Q

r r p+1r p+1r

hr Q r Q i

x r x p+1r

z z z z

r =1 r =2

p2 49

and from the homogeneity we conclude that for p we have

p1 p

X X

Q Q Q Q

r r p+1r p+1r

hr Q r Q i

x r x p+1r

z z z z

r =2 r =1

Claim For p Q is of degree p in z

Pro of of Claim For p equation reduces to

so this is certainly true for p If it is assumed to hold for Q

Q it then follows from equation that it also holds for Q

p1 p

Hence our claim follows by induction

r 2

Now let L z x denote the co ecient of z in Q for r where x

r r

x x From equation we see that Q L Furthermore

3 m 2 2

the co ecient of z in equation must b e

p L hr L r L i

p x r x p+1r

r =2

In particular if L x L x we see that

r r

p L hr L r L i

p x r x p+1r

r =2

We write L x x Ax for some symmetric matrix A

Claim For p we have

p t p1

L x x A x

Pro of of Claim This holds by denition for p If it is assumed

to hold for L L then we have

2 p1

r r 1

r L x A x

x r

for r p Hence from equation we see that

p t p1

L x x A x

and our claim follows by induction

Since d is the degree of F it follows that

t p

x A x 50

for p d Since A is assumed to b e symmetric this implies that A is

nilp otent so traceA Thus we get at the p oint

F traceA

z z

The origin was arbitrarily chosen among the nonsingular p oints of F

so it follows that

F p

for every nonsingular p oint p of F Since F is a horizontally conformal

polynomial its set of critical p oints is nowhere dense Hence F is

harmonic

Following Theorem and Theorem the study of globally dened

m n

harmonic morphisms R R with n is reduced to the study of

horizontally conformal p olynomials of degree

m n

Corollary If n m n and R R is a non

constant harmonic morphism then is an orthogonal projection

fol lowed by a homothetic isomorphism

Proof We assume for simplicity that Since n n

it follows that n and m n Hence is linear by The

orem Cho ose an orthonormal basis ff f g of the horizontal

1 n

space ker so that f e for i n where fe e g

i i 1 n

is the canonical base of R and the dilation of Then is easily

seen to b e the orthogonal pro jection onto the horizontal space followed

by a homothetic isomorphism

Example If m n then any harmonic morphism R

R is p olynomial of degree no more than We have in this case the

following result which we state without pro of

m n

Theorem If F R R is a polynomial harmonic mor

phism of degree then there is an orthonormal basis for R such that

F F is either homogeneous of degree or of the form

x x x x Qx x

1 m 1 n n+1 m

for some R and some polynomial harmonic morphism

mn n

Q R R

homogeneous of degree

Theorem together with the results of Section give a complete

m n

classication of all harmonic morphisms R R with m n 51

The Classication of Ou

In this section we study harmonic morphisms dened by homoge

neous p olynomials of degree Due to a connection with the represen

tation theory of Cliord algebras we obtain a complete classication

of these In our presentation of this classication we follow the work

of Ou and Wo o d in and Ou in We also derive some results

needed in the next chapter

m n

Denition A map R R is said to b e quadratic if

each of its comp onents are quadratic forms on R The set of all qua

m n

dratic harmonic morphisms R R will b e denoted by H m n

Two quadratic maps are said to b e domainequivalent if there is

an A O R such that A and biequivalent if there is a

m n

B O R and a C O R such that C B

We see that domainequivalence amounts to an orthonormal change

of co ordinates of R and biequivalence to an orthonormal change of

co ordinates in b oth the domain and the codomain

m n

If R R is a quadratic map we may write

i m

x hA x xi x R

for i n and some symmetric A End R Henceforth we

make no distinction b etween EndR and the set of m mmatrices

over R thus refering to A as the ith component matrix of We also

equip EndR with the standard inner pro duct

traceA B hA B i

m n

Denition A quadratic map R R with comp onent

matrices A i n is said to b e separable if it is p ossible to write

R as a direct sum of nontrivial subspaces invariant under A for

every i Otherwise is said to b e nonseparable

From Theorem we may now derive the following equations

m n

Prop osition A quadratic map R R with com

ponent matrices A i n is a harmonic morphism if and only

2 2

a A A for al l ij

i j

b A A A A for al l i j and

i j j i

c traceA for al l i

i 52

Proof For every i we have

i R i

grad x A x and traceA

i i

Thus c is immediate and a and b follow from Example Observe

that for b A A is not symmetric in general

i j

2 2

Example For t R we see that R R given by

2 2

x y x cos t y cos t xy sin t

2 2

x sin t y sin t xy cos t

has comp onent matrices

sin t cos t cos t sin t

cos t sin t sin t cos t

They do satisfy the equations of Prop osition so is a quadratic

harmonic morphism

Denition For manifolds M N and maps f M R and

n n

g N R we dene their direct sum as the map f g M N R

by f g x y f x g y where M N is given the usual pro duct

structure

It follows directly from Prop osition that a direct sum of qua

dratic harmonic morphisms gives new quadratic harmonic morphisms

The direct sum construction can of course b e applied to more than two

terms

Lemma The Rank Lemma Let H m n have

component matrices A A Then

1 n

al l the component matrices have the same rank which is an even

number

al l the component matrices have the same eigenvalues and

the map is domainequivalent to a quadratic harmonic mor

phism with component matrices F F with

1 n

B D

A A

D B

F i n F

i 1

where D is a diagonal matrix with only positive diagonal entries

and B matrices satisfying

t 2

B D DB B B D for al l i n and

i i i

t t

B B B B for al l i j n i j

j i

i j

The map is said to be of the normal form 53

The pro of of this lemma is a simple application of the equations

in Prop osition using wellknown facts ab out symmetric matrices

We omit this and refer to page of

Example The Hopf p olynomials of Example are all

quadratic harmonic morphisms

2 d d+1

R R

for d or They are all of normal form in the canonical basis

2d 2 2

of R For d x y x y xy has comp onent matrices

For d the corresp onding Hopf p olynomials are given by

2 2 2 2

x x x x x x x x

1 2 3 4

x x x x x x x x

1 3 2 4 1 4 2 3

with comp onent matrices

C B C B C B

A A A

The next one with d is given by

2 2 2 2 2 2 2 2

x x x x x x x x x x

1 8

8 7 6 5 4 3 2 1

x x x x x x x x

1 5 2 6 3 7 4 8

x x x x x x x x

1 6 2 5 3 8 4 7

x x x x x x x x

1 7 2 8 3 5 4 6

x x x x x x x x

1 8 2 7 3 6 4 5

so it is also of the normal form It is easy to see that this is also the

case for d

Denition If H m n then the common rank of its

comp onent matrices is called the Qr ank of If the Qrank of is m

then is said to b e Qnonsingular If H m n is Qnonsingular

and the p ositive eigenvalues of its comp onent matrices are the same

then is said to b e umbilical The set of all umbilical elements of

H m n with p ositive eigenvalue is denoted by H m n

Example We see that the Hopf p olynomials are all umbilical

with p ositive eigenvalue and so are the maps of Example Note 54

n n m m

2 1

R are umbilical quadratic R and R that if R

2 1

harmonic morphisms with p ositive eigenvalue then for R

m +m n

1 2

R R

1 2

will in general not b e umbilical We shall later see that this is the only

way to construct nonumbilical quadratic harmonic morphisms

Corollary Any quadratic harmonic morphism is the

composition of an orthogonal projection fol lowed by a Qnonsingular

quadratic harmonic morphism from an evendimensional space

Proof The statement is clearly true for a quadratic harmonic

morphism of the normal form The general case follows from Lemma

We may compare Corollary with a result of Baird and Wo o d

see Theorem of stating that any nonconstant harmonic mor

3 2 3

phism R R is the comp osition of an orthonormal pro jection R

R followed by a weakly conformal map By Corollary we see that

it is enough to study Qnonsingular quadratic harmonic morphisms

from evendimensional spaces

Corollary If H m n is umbilical then is domain

equivalent to a H m n given by

x hP x xi hP x xi

1 n

for some real constant and P S y mR satisfying

P P P P I for al l i j

i j j i ij m

where I is the identity endomorphism of R

Proof This follows immediately from Lemma

Denition An ntuple P P of symmetric endomor

1 n

phisms on R satisfying

P P P P I

i j j i ij m

for all i j n is called an ndimensional Cliord system on

m m

R The set of all ndimensional Cliord systems on R is denoted by

C m n

We note that from the last denition the only p ossible eigenvalues

for any P is Since P with i j denes an isomorphism b etween

i j

the eigenspaces E and E of P these spaces have the same dimen

1 1 i

sion From the symmetry of P we conclude that m dimE

i 1

Hence C m n for m o dd It also follows that traceP for all

i 55

i so that every Cliord system denes a quadratic umbilical harmonic

morphism with p ositive eigenvalue

Corollary Up to a homothetic change of coordinates of

R and domainequivalence every umbilical quadratic harmonic mor

phism is given by a Cliord system as in Corol lary In particular

if m is odd then there are no umbilical quadratic harmonic morphisms

m n

R R

We now fo cus our attention on Cliord systems

Denition Two Cliord systems P P and Q

1 n 1

2 m

Q on R are said to b e algebraical ly equivalent if there exists an

A O m such that

A P A Q

i i

for all i n They are said to b e geometrical ly equivalent if there

exists a B O span fP P g such that B P B P and

R 1 n 1 n

Q Q are algebraically equivalent

1 n

It follows see Theorem that two Cliord systems are

algebraically geometrically equivalent if and only if the corresp onding

quadratic harmonic morphisms are domainequivalent biequivalent

Denition If P P C m n and Q Q

1 n 1 1 n

2m 2m

1 2

C m n are two ndimensional Cliord systems on R and R

resp ectively then their direct sum is the ndimensional Cliord sys

2 (m +m )

1 2

tem on R given by P Q P Q A Cliord system

1 1 n n

2 m

P P on R is said to b e irreducible if it is not p ossible to write

1 n

R as a direct sum of nontrivial subspaces invariant under all P

It is easy to see that a Cliord system is irreducible if and only if it

is not algebraically equivalent to a direct sum of two Cliord systems

Furthermore irreducible Cliord systems corresp ond to nonseparable

quadratic harmonic morphisms

As noted in there is a natural connection b etween Cliord

systems and the representation of the Cliord algebras C From this

connection the following may b e deduced

Theorem The fol lowing facts hold for Cliord systems

Every Cliord system is algebraical ly equivalent to a direct sum

of irreducible Cliord systems

Irreducible Cliord systems P P C m n exist pre

1 n

cisely for the values of n and m mn listed in Table 56

For n mod al l Cliord systems in C mn n are alge

braical ly equivalent

For n mod there are in C mn n two equivalence

classes under algebraic equivalence and one under geometrical

equivalence

n n

mn mn

Table

From Corollary and Theorem we may now prove existence

of umbilical quadratic harmonic morphisms

Theorem For any n N there exist nonseparable um

2m n

bilical quadratic harmonic morphisms R R for exactly the values

of m n mn n listed in Table Other umbilical quadratic har

n 2 k m n

monic morphisms into R exist exactly in the cases R R when

m n mn n is contained in Table and k N

Proof The statement follows directly from the fact that

by Corollary any umbilical quadratic harmonic morphism cor

resp onds to a Cliord system which is algebraically equivalent to a

direct sum of irreducible Cliord systems

Example The Hopf p olynomials are clearly given by Cliord

systems in C mn n with n or Since every Cliord

system in C mn n is irreducible the Hopf p olynomials are all non

separable

In order to give a complete characterization of quadratic harmonic

morphisms we next prove that umbilical quadratic harmonic morphisms

are the building blo cks for the general case

2 m n

Lemma The Splitting Lemma If R R is

a Qnonsingular quadratic harmonic morphism then is domain

equivalent to a direct sum of umbilical quadratic harmonic morphisms

Proof If is umbilical we are done If not let

1 m

b e the p ositive eigenvalues of the comp onent matrices of counted by

for k l m Then is multiplicity with

1 2 k l

by Lemma domainequivalent to where is of the normal form

with D diagonal and diagonal entries From the fact that

1 m

DB B D we see that

i i

i 57

k m k

for some b GLR c GLR i n Using the isometry

i i

B C

H and that we see that is domainequivalent to

1 2

where

2 k n 2 (mk ) n

R R and R R

1 2

By Prop osition the maps and are b oth quadratic harmonic

1 2

morphisms and is umbilical by construction Rep eating the pro ce

dure completes the pro of in a nite numb er of steps

We may now give the characterization theorem for Qnonsingular

quadratic harmonic morphisms By Corollary this also covers the

general case

Theorem For n N Qnonsingular quadratic har

monic morphisms with values in R exist precisely in the cases

2k m( n) n

R R

where k N Each such map is domainequivalent to a direct sum of

the kind

1 1 k k

mn n given by an irreducible Cliord system for some H

Proof This follows directly from Theorem and Lemma

Example Fix k N By Theorem any Qnon

singular quadratic harmonic morphism

2k m( n) n

R R

is up to domainequivalence of the form

1 1 k k

k 1

for some ktuple R and some H mn n

1 k i

i k given by irreducible Cliord systems If n mo d

it follows from the prop erties of Cliord systems that every two 58

elements of H mn n are domainequivalent Hence up to domain

equivalence is in this case of the form

1 k

where H mn n is given by irreducible Cliord systems In

particular we see that for a xed ktuple R any two

1 k

quadratic harmonic morphisms with these eigenvalues are domain

equivalent

If n mo d the situation is more complicated since in this

case there are two algebraic equivalence classes in C mn n But as

noted by Ou see Corollary two nonequivalent Cliord sys

tems will only dier by a sign of one say of the last comp onent Hence

there are in this case two domainequivalence classes in H mn n

and two nonequivalent quadratic harmonic morphisms in this set will

dier only by a sign of one say the last of their comp onents This im

k k

plies that for a xed ktuple R we get p ossibilities of

1 k

constructing our Qnonsingular quadratic harmonic morphism of the

form

1 1 k k

mn n Half of these p ossibilities can b e obtained from for H

the other by an orthonormal change of co ordinates in R as one easily

k 1

checks Hence there are in this case biequivalent classes In

particular for k and n N there is just one biequivalent class in

mn n H

Polynomial Harmonic Morphisms of Higher Degree

So far we have not seen any explicit examples of p olynomial har

monic morphisms of degree higher than except when the target man

ifold is C Of course by comp osing quadratic harmonic morphisms we

may construct examples of even degree but it was an op en question

for several years whether there exist p olynomial harmonic morphisms

of higher degree that do not arise in this way For instance do there

exist p olynomial harmonic morphisms of o dd degree This question

has recently b een given an armative answer in some cases in by a

surprisingly simple construction We shall in this section describ e this

metho d and show how it provides a multitude of examples

Denition A multilinear map

p p n

R R R 59

is said to b e normpreserving if

jx x j jx j jx j

1 k 1 k

p p

R for every x x R

1 k

Example It is easily seen that the following maps are

normpreserving

2 3

C C z z z z z z iz z z C

1 2 3 1 2 2 2 2 3

and

C C z z z z

1 2 3 4

z z z z z z z z z z C

1 2 1 3 1 4 4 1 4 4

Example For any k N and d or let

d d d

R R R x x x x x x

1 k 1 k 2 k 1 k

b e the multiplication of real complex quaternionic or Cayley num

b ers Then is normpreserving Actually these are the only p ossible

dimensions for the existence of a multilinear normpreserving map

n n n

R R R

n1 n1

with k For if we x x x S S then the

3 k

n n n n

resulting map R R R will turn R into a normed division

algebra hence we must have n or by a classical result of

Hurwitz see We shall make use of this fact in the pro of of the

following theorem

Theorem A normpreserving multilinear map

p p n

R R R

is a harmonic morphism if and only if p p n or

1 k

Proof If p p n then is a harmonic morphism

1 k

by Theorem and Example since is a homothetic linear trans

formation in each variable separately

p p n

Conversely assume that R R R is a multilinear

normpreserving harmonic morphism Since is normpreserving it is

clear that

p n

i 60

for i n If is the dilation of we see that for x x x

1 k

p 1 p 1

S S we have

n x x tracehd d i p p p

1 k x x 1 2 k

For a function f R R we get from equation together with

Example

n p

2 R R

x x f x x f x

1 k 1 k i i

i=1

where x x x If we cho ose f y jy j so that

i 1 i k

R 2 2 p 1 p 1

f y njy j jy j then for x x S S

1 k

we have

f x p

i i i

From equations and it then follows that

p p p

1 2 k

and comparing with equation we obtain p p p

1 2 k

n The theorem now follows from the previously mentioned result of

Hurwitz

Using Ktheory Tang has recently improved Theorem and ob

tained the following

p p n

Theorem If the map F R R R is

multilinear and nonsingular ie F x x implies that x

1 k i

for some i then F is a harmonic morphism if and only if p

p n or

From Theorem we may now for any k N and n or

construct p olynomial homogeneous harmonic morphisms of degree k

n n n

R R R

Example For the nonhomogeneous case we have the fol

lowing construction For Riemannian manifolds M M and a fam

1 k

ily M R i k of harmonic morphisms their direct

i i

sum

M M R

1 k 1 k

is given by

x x x x

1 k 1 k 1 1 k k 61

for x x M M Since this is a harmonic morphism in

1 k 1 k

each variable separately it is a harmonic morphism by Example

p n

Now assume that A R R is a homogeneous p olynomial

harmonic morphism for i k Dene the map

p p n

F R R R

as the direct sum of the A s

F x x A x A x

1 k 1 1 k k

The map F is a harmonic morphism and if the A s are not all of the

same degree then F will b e nonhomogeneous

After Example was pro duced it was p ointed out by Ou that a

sp ecial case of this construction has app eared in But since this

is written in Chinese it is our hop e of not b eing accused of plagiarism

Example motivates the following denition

m n

Denition A harmonic morphism R R is said to

b e separable if up to isometries can b e written as a direct sum of

harmonic morphisms from spaces of strictly lower dimensions Other

wise is said to b e nonseparable

The map F constructed in Example is separable and by The

orem any p olynomial harmonic morphism of degree is either ho

mogeneous up to an additive constant or separable We make the

following conjecture

m n

Conjecture If R R is a nonseparable polyno

mial harmonic morphism then up to an additive constant is homo

geneous 62

CHAPTER

Applications

In this chapter we present a new way of proving two wellknown

theorems of Eells and Yiu The rst states that the Hopf p olynomials

are essentially the only homogeneous p olynomial harmonic morphisms

preserving spheres The second states that these are essentially the

only harmonic homogeneous p olynomials that restricts to harmonic

morphisms b etween spheres For this we use results from Chapter

together with some imp ortant results of algebraic and dierential

top ology This is then used to derive information on the singularities

of general harmonic morphisms

The Theorems of Eells and Yiu

The results needed from algebraic top ology originate in the follow

ing remarkable result of W Browder see Spanier for the deni

tions

Theorem Let F be a connected polyhedron and p S

B a weak bration over B with bre F If B is not a single point then

1 3 7

F must be homotopic to S S or S

Timourian used Theorem to obtain the following result

Lemma more suitable for our purp oses

Theorem If m n and T is a homotopy msphere

m n

and T S is a bre bund le with compact m ndimensional

bres then m n or

Note that these are exactly the dimensions of the Hopf brations

of Example with m n It is well known that all the Hopf bra

tions are bre bundle maps and they are indeed surjective submersions

b etween compact manifolds The following result is due to Ehresmann