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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Subseries: Mathematisches lnstitut der Universitiit und Max-Planck-lnstitutfur Mathematik, Bonn Adviser: E Hirzebruch
Jurgen Jost
Harmonic Maps Between surfaces (with a Special Chapter on Conformal Mappings)
Springer-Verlag Berlin Heidelberg New York Tokyo I984
Author
J~irgen Jost Mathematisches Institut der Universit~t Wegelerstr. 10, 5 3 0 0 Bonn, Federal Republic of Germany
A M S Subject Classification (1980): 58 E20; 30 C 70, 32 G 15, 35J 60 ISBN 3 - 5 4 0 4 3 3 3 9 - 9 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0 - 3 8 7 4 3 3 3 9 - 9 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright.All rights are reserved,whetherthe whole or part of the material is concerned,specificallythose of translation,reprinting, re-use of illustrations,broadcasting, reproduction by photocopyingmachineor similar means,and storage in data banks. Under § 54 of the GermanCopyright Law where copies are madefor other than private use, a fee is payableto "VerwertungsgesellschaftWort", Munich. © by Springer-VerlagBerlin Heidelberg1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 214613140-543210
Dedicated
to the
memory
of
Dieter
Kieven
P R E F A C E
The p u r p o s e
of these L e c t u r e
give a fairly harmonic
complete
maps b e t w e e n
Notes
surfaces.
also serve as an i n t r o d u c t i o n ral;
therefore,
dimensional
whenever
results
and I try to give
should consult
On the other hand,
account
I want
of the results these notes
to on
should
in gene-
I p o i n t out w h i c h of the two -
to h i g h e r
some references
the several
On one hand,
account
to the theory of h a r m o n i c maps
appropriate,
pertain
For a more c o m p l e t e
is twofold.
and self h c o n t a i n e d
dimensions
and w h i c h
do not,
and an idea of the r e s p e c t i v e
in this direction, excellent
however,
survey a r t i c l e s
proof.
the r e a d e r
of Eells
and Le-
maire. An e s s e n t i a l
a i m of this book
the i n t e r p l a y particular
of d i f f e r e n t
the calculus
differential
geometry,
the c o n c e p t
of this book
In particular,
Nevertheless, cations
I believe
tion of u n r e l a t e d
proved.
until
thorough way
differential
conformal
Thus,
unified
treat-
contains
several
simplifi-
available
the s e q u e n c e
of the three are
in
as a m e r e e n u m e r a -
and m a n y d i f f e r e n t
mappings
than in the e x i s t i n g
equations,
n o r desirable.
On the contrary,
the results
in
to the view of a mere
is n o t i n t e n d e d
order,
and
maps,
analysis.
to the p r e s e n t a t i o n s
This b o o k
a logical
In p a r t i c u l a r ,
opposed
treatment
compared
results.
ters also r e f l e c t s
partial
that a c o m p l e t e l y
possible
that this
literature.
be constructed,
I think
is n e i t h e r
of m e t h o d s
theory of h a r m o n i c
topology, and c o m p l e x
is s t r o n g l y
and u n i f i c a t i o n s
the e x i s t i n g
in the
of variations, algebraic
specialist.
m e n t of the topic
is to show the v a r i e t y
fields
final
of the chap-
tools have
chapters
to
can be
used in a much m o r e
literature.
An outline
of the con-
tents n o w follows. After
giving
of h a r m o n i c in c h a p t e r discs
an a c c o u n t of the h i s t o r y maps
some
on surfaces,
are no c o n j u g a t e Moreover, controlled Chapter
points
geometric
points
Christoffel
"Multiple
contains
is r o u g h l y
then there are
w e show the e x i s t e n c e symbols,
following
c o n f o r m a l mappings.
a mistake.
These
I, we start concern c o n v e x
that if on a disc there
also no cut points.
of local
Integrals...",
the d e f i n i t i o n
of view in c h a p t e r
considerations.
a n d the result
3 deals w i t h
in Morrey, proof
from several
2 with
and p r e s e n t i n g
coordinates
Jost-Karcher
[JK1]
We first p r o v e T h e o r e m
Springer,
The d i f f i c u l t y
with curvature
which
1966, leads
9.3
since M o r r e y ' s to this e r r o r
is
VI
overcome
by minimizing
H 2I w h i c h
space
vertheless
conclude
sired
properties
11).
Furthermore,
morphism ever,
and
will
that
we
4, w e
the boundary
first
general which
a - priori We
maximum
estimates
then attack
compact hard
to s e e
homotopy group
of the
obtain [LI],
[L2].
image
In c h a p t e r
maps
and prove
Kaul
[J~KI]
problem
problem
values,
results
detail
lent harmonic
unit 6 and
mappings
boundary
disc
between
maps,
of
disc.
results
results
and we
thus
of Lemaire we
can al-
classes
for
to a 2 of harmonic
[Ht]
and J~ger-
case
of m a p s
be-
surfaces.
and since
f o r the c a s e w h e r e
latter
These
estimates map
the existence problem,
apply
in p a r with
a
c h a p t e r 3 to p a s s 9. T h e
results
and employ
several
of harmonic
7.
is t a k e n
diffeomorphisms
if the b o u n d a r y
homeomorphically
result
of chapter
we
for u n i v a -
composed
in c h a p t e r [JKI]
can
of
Heinz.
the D i r i c h l e t the d o m a i n
domain
7 where
from below
a conformal
on Jost-Karcher
assumption
of c h a p t e r
determinant
to a n a r b i t r a r y
o f E.
This
This
harmonic, w e c a n use the r e s u l t o f
8, w e p r o v e of
of t h e
functional
7 are based
ideas
in the p l a n e .
the help the
one is a g a i n
In c h a p t e r
convex
for
to c o n f o r m a l
solutions
the
unit
removed with
estimates
important
out of a
homotopy
surfaces.
disc
chapter
fall
argument,
the
map.
between
i t is n o t
of uniqueness
C 1'e - a - p r i o r i - e s t i m a t e s
from the
can
second
homotopy
is the
harmonic
again,
of H a r t m a n n
in s o m e m o r e
gives
maps
is h o m e o m o r p h i c
the q u e s t i o n
of a
the h a r m o n i c
however,
replacement
the domain
ticular
that
and Le-
also
theorems
6, w e p r o v e
then be
of
If t h e
In c h a p t e r
prove
proof
minimizing
in two d i f f e r e n t
the c o r r e s p o n d i n g
how-
to H i l d e -
to C o u r a n t
Lemma
off.
if the i m a g e
5, w e d e a l w i t h
due
for h a r m o n i c
existence
by a careful
for the c a s e
The
cannot happen,
fundamental
and then examine
closed
due
of continuity
splits
this
estimates,
a result
dimensions.
existence
vanishes, o f the
4 and
diffeo-
consists o f a c o m b i n a t i o n
limit of an energy
boundary
problem
and a lemma
two
if a s p h e r e
the D i r i c h l e t
sphere.
tween
only
(A-priori
the Courant-Lebesgue
Furthermore,
nonconstant
in
is a g l o b a l
the d e -
chapters).
convex ball,
the m o d u l u s
Using
a new proof
so s o l v e
for
t h a t the
class
in c h a p t e r s
idea
this m a p
the D i r i c h l e t
principle
the general
surfaces.
a similar
[ H K W 3]. O u r p r o o f
is o n l y v a l i d
Sobolev
map with
could expect
in s o m e
of the
so that we can ne-
is a c o n f o r m a l
that
in later
solve
lie
brandt - Kaul - Widman rather
as o n e
obtained
values
besgue
encounter prove
subclass
to the p r o b l e m ,
the m i n i m u m
shall
as r e g u l a r
in a r e s t r i c t e d
adapted
(we s h a l l
only be
In c h a p t e r
energy
is s u i t a b l y
from
onto [J3]
values
a convex and uses
map
as the
curve inside in p a r t i c u l a r
a
VIi
We can also use the proofs
of Thms.
a - priori - e s t i m a t e s
4.1 and
8.1 in chapter
to p r o v i d e
9, using
non - v a r i a t i o n a l
Leray -Schauder
degree
theory. We then apply T h e o r e m monic
coordinates
8.1 in c h a p t e r
on a r b i t r a r y
These
Karcher
[JK1].
perties
and can be u s e d to prove
coordinates
nic maps b e t w e e n
surfaces
jectivity
once
Theorem
radii,
The
final
chapter
surfaces.
First,
m o n i c maps
gives we give
the a n a l y t i c
ature
is h a r m o n i c , a s
Gauss
curvature.
A mong
the o m i s s i o n s
well
insights
into
or c o n v e r s a t i o n s Schoen
of this
Yau.
that w e
Finally,
comments
on m y m a n u s c r i p t
that the
constant
mean
curv-
on the e x p l i c i t
between manifolds [EW 2],
can c o n t r i b u t e
To H e r m a n n Karcher,
[EW 3],
anything
I owe many
of the field w h i c h he g e n e r o u s l y
I benefitted
much
Bob Gulliver,
advice
my research
from c o l l a b o r a t i o n
Luc Lemaire,
Rick
joint w o r k w i t h him),
But most of all,
for his continous
of Bonn.
stating
to [L I],
11 represents
and for s u p p o r t i n g
the means
of a w e l l
in 3 - space of c o n s t a n t
maps
the reader
aspects
Chapter
through
maps b e t w e e n
area.
Furthermore,
John W o o d and S h i n g - T u n g
with
to show that the
and T r o m b a of har-
book are results
of h a r m o n i c
w i t h J i m Eells,
Stefan H i l d e b r a n d t
due to
c l o s e d surfaces,
space w i t h
surfaces
to several persons.
(in particular,
many years,
as i m m e r s e d
the g e o m e t r i c
c o m m u n i c a t e d to me.
between
of Earle - Eells
since w e do not feel
is i n d e b t e d
the
surfaces,
p r o o f of Eells - W o o d
of E u c l i d e a n
We refer
new to the p r e s e n t a t i o n My w o r k
and in-
theory.
of the p r e s e n t
metrics.
[EL 3] instead,
bounds
of d i f f e o m o r p h i s m s
argument
the T h e o r e m of R u h - V i l m s
and c l a s s i f i c a t i o n
canonical
closed
of h a r m o n i c
mappings
some a p p l i c a t i o n s
we discuss
construction
pro-
for h a r m o -
is known.
in the class
replacement
concerning
Gauss m a p of a s u b m a n i f o l d
with
regularity
estimates
11 w h e r e we prove
between
some a p p l i c a t i o n s
to T e i c h m O l l e r
Furthermore,
of har-
to Jost -
diffeomorphism.
result of Kneser
a n d then we give
in c h a p t e r
energy
delicate
according
only on c u r v a t u r e
of c o n t i n u i t y
diffeomorphisms
a rather
is a h a r m o n i c
k nown
depending
[JS]. We m i n i m i z e
a n d then apply
the e x i s t e n c e
best p o s s i b l e
C 2'~ - a - priori
the m o d u l u s
of h a r m o n i c
Jost - S c h o e n
limit
possess
8.1 w i l l again be a p p l i e d
existence
10 to prove
discs on a surface
I am indebted
and e n c o u r a g e m e n t
to over
in every p o s s i b l e w a y
of the S o n d e r f o r s c h u n g s b e r e i c h
72 at the U n i v e r s i t y
I am grateful
for some useful
to A l f r e d
and to M o n i k a
great care and patience.
Baldes
Zimmermann
for typing it
Table
of contents I
I. I n t r o d u c t i o n 1.1.
A short history
1.2.
The
1.3.
Definition
1.4.
concept
2.
Physical
1.6.
Some
~ometric
2.2.
Convexity
2.3.
Uniqueness
2.4.
Remark:
Curvature
2.6.
Local
of
of g e o d e s i c s
the s q u a r e d
of geodesic
arcs
distance
arcs
. . . . . . . .
and conjugate
discs
10 10
14 15
curves . . . . . . . . . . . . . . . .
15
curvature
o f 2.3.
13
. .....
with
analogue
. . . . . . . .
controlled
Christoffel
mappings
16
18 3.1
surfaces
concerning
Proof
Lemma
of Theorem
3.7.
Uniqueness
3.8.
Applications
3.9.
The Hartman-Wintner
3.1
of conformal
~eorems
3.1
A maximum
The Dirichlet
principle
to p l a n e
domains
.....
18 19
. . . . . . . . . . . . . . . .
21
. . . . . . . . . .
33
. . . . . . . . . . . . . . . .
34
Lemma . . . . . . . . . . . . . . . . .
35
for h a r m o n i c
4.2.
representations
. . . . . . . . . . . . . . . .
representations
of T h e o r e m
4.1.
conformal
homeomorphic
The Courant-Lebesgue - 3.6.
9
points12
. . . . . . . . . . . . . . . . . . . . . . . . . .
compact
3
of harmonic
function . . . . . . . . .
in c o n v e x
dimensional
of p a r a l l e l
S t a t e m e n t o f Thm.
4. E x i s t e n c e
and terminology
1 2
12
existence
coordinates
symbols
3.3.
concept
. . . . . . . . . . . . . . . . . . .
about notation
The higher
2.5.
of
from the
considerations
Convexity,
3.2.
. . . . . . . . .
. . . . . . . . . . . . . . . .
arising
significance
remarks
2.1.
3. C o n f o r m a l
maps
problems
principles
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
I .5.
3.1.
of harmonic
Mathematical maps
of variational
of geodesics
maps between
for energy
problem,
if the
minimizing image
38
surfaces maps . . . . . . .
is c o n t a i n e d
38
in a c o n v e x
disc . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
4.3.
Remarks
42
4.4.
The
Theorem
4.5.
The
Dirichlet
problem,
Two
different
solutions,
stant
about
the h i g h e r
of L e m a i r e
-dimensional
situation
and Sacks-Uhlenbeck if the i m a g e if
......
. . . . . . . . .
is h o m e o m o r p h i c
the b o u n d a r y
values
are n o n c o n -
. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.
Nonexistence
4.7.
Existence
for c o n s t a n t
results
boundary
in a r b i t r a r y
43
to S 2
46
values
. . . . . . . . .
50
dimensions
. . . . . . . . .
52
IX
5. U n i q u e n e s s 5.1.
Composition
5.2.
The
5.3.
of harmonic
uniqueness
Uniqueness positive
5.4.
for the
Uniqueness
6. A - p r i o r i
problem
for closed
....
54
. . . . . . . . .
54
if t h e
image has
non-
solutions,
if the i m a g e
62 has
curvature . . . . . . . . . . . . . . . . . . . and nonuniqueness
for harmonic
maps
63
between 65
estimates
65
Cc~position
6.2.
A maximum
6.3.
Interior
modulus
6.4.
Interior
estimates
6.5.
Boundary
continuity
of h a r m o n i c
principle
of continuity
conformal
maps
.....
65 66
. . . . . . . . . . . . . .
68
f o r the e n e r g y . . . . . . . . . . . . .
68
. . . . . . . . . . . . . . . . . . .
Interior
C I -estimates
6.7.
Interior
C 1'e - e s t i m a t e s
6.8.
C I - and C 1'~-estimates
estimates
maps with
. . . . . . . . . . . . . . . . . . . .
6.6.
7.1.
functions
. . . . . . . . . . . . . . . . . . . .
6.1.
harmonic
convex
and Kaul
surfaces . . . . . . . . . . . . . . . . . . . . . .
C 1'e
7. A - p r i o r i
maps with of J ~ g e r
Dirichlet
results
Uniqueness closed
theorem
curvature
nonpositive 5.5.
54
theorems
69
. . . . . . . . . . . . . . . . . .
72
. . . . . . . . . . . . . . . . .
75
at t h e b o u n d a r y . . . . . . . . . .
75
from below
for t h e
functional
determinant
of
diffeomorphisms
A Harnack
77
inequality
o f E. H e i n z
. . . . . . . . . . . . .
77
7.2.
Interior
estimates
. . . . . . . . . . . . . . . . . . . .
78
7.3.
Boundary
estimates
. . . . . . . . . . . . . . . . . . . .
82
7.4.
Discussion
8. T h e e x i s t e n c e
of t h e
of h a r m o n i c
situation
in h i g h e r
diffeomorphisms
dimensions
which
solve
.....
84
a Dirichlet 86
problem 8.1.
Proof
o f the e x i s t e n c e
tained
in a convex
8.2.
Approximation
8.3.
Remarks: Theorem
9.
Plane
in c a s e
and bounded
the image
is c o n curve.
. .86
arguments . . . . . . . . . . . . . . . . . .
88
domains,
necessity
by a convex
of the hypotheses
of
8.1 . . . . . . . . . . . . . . . . . . . . . . . .
C I'~ - a - p r i o r i existence
theorem
ball
estimates
for
arbitrary
domains.
Non - variational
proofs
90
9.1.
C I'~ - e s t i m a t e s
9.2.
Estimates
9.3.
A non-variational
surfaces
90
for
on arbitrary the f u n c t i o n a l
surfaces determinant
. . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . proof
of T h e o r e m
90
on arbitrary
4.1 . . . . . . . . . .
92 93
9.4.
10.
11.
A non -variational
Harmonic
Existence
10.2.
C 2"~ -estimates
10.3.
Bounds
10.4.
Higher
10.5.
C 2'~-
10.6.
Higher
The
diffeomorphisms 11.1)
11.3.
Extension
11.4.
Remarks
Applications
12.3.
the
of Kneser's
Proof
of T h e o r e m
Contractibility
. . . . . . . . . . .
between closed
surfaces
. . . . . . . . . . . . . . . . in h i g h e r
between
dimensions
....
Tromba's
proof
12.5.
The
12.6.
Harmonic
Gauss
12.7.
Surfaces
of
of
103 104 104 105
106
106 106 111 111
112
surfaces
certain
harmonic
Theorem
. . . . . . . . . . . . . . . .
112
12.1 . . . . . . . . . . . . . . . . . .
113
of
Teichm~ller
maps
98 100
surfaces
space
and
and
an a n a l y t i c
the
diffeomor-
. . . . . . . . . . . . . . . . . . . . . . .
12.4.
References
8.1
maps
of
proof
approach
maps
between
situation
of harmonic
group
. . . . . . . .
11.1 . . . . . . . . . . . . . . . . . .
of Theorem
Holomorphicity
phism
coordinates
. . . . . . . . . . . . . . . . . . . . .
of T h e o r e m
about
coordinates
diffeomorphisms
(Theorem
....
. . . . . . . .
Conformal
95
m a p s 98
maps . . . . . . . . . . . .
of harmonic
Harmonic
for harmonic
C 1'e - e s t i m a t e s
symbols.
of harmonic
of harmonic
Proof
estimates
coordinates
for harmonic
regularity
8.1 . . . . . . . . . .
coordinates.
harmonic
Christoffel
regularity
11.2.
12.2.
the
for
estimates
existence
12.1.
of h a r m o n i c
on
of T h e o r e m
C 2'~ - a-priori
10.1.
11.1.
12.
coordinates.
proof
that
TeichmOller
space
Gerstenhaber-Rauch maps
constant
and
Bernstein
Gauss
is a
cell . . . . .
. . . . . . . . . . . theorems
curvature
in
. . . . . . .
3- space .....
116 119 123 124 124
127
GROUP
EPIMORPHISMS
AND
PRESERVING
THE
If
one
uses
various
natural
topological
for
certain
class
i.
Definition
its
commutator
To
define
commutator
class
under
is
of
class,
element, subgroup.
if t An
recall
that
subgroups of
if
each the
the
derived
to to
a group
Because
It
subgroup H
perfect
The
G.
a
'~
the
have
being
obtain
this
an
plus-
algebraic
viewpoint
P
is
perfect
homomorphic
radical
series
here of
a given
is
also
is
if
image
group
PG is
a
equal of
is G.
of
to
by
Thus
the
G~
¢(PG)
view
is
to
a
which
PG
its
must
~ P(tG) the
under commutators,
class
admits
be
because,
as
therefore
closed
generated
functorial
then
G
evidently
generate.
construction
approach
of
H's
all
: G ÷ H is a homomorphism~
transfinite
sequence
back,
Central
K-theory,
excision)
so
perfect
alternative
a fibre
pendulum
algebraic to
image of a perfect group is again perfect.
subgroup
invariant
as
matters.
[P,P].
because the
the
to
obstruction
homomorphisms.
automorphisms
union, so
swing these
a commutator,
a maximum
the
we
subgroup
is
Tbs closed
i.2
(such
the homomorphic
i.i
then
formulations
group
the
approach
example,
discussing of
Berrick
(for
Here
setting
J.
plus-construction
problems
constructive).
group
the
key
RADICALS,
PLUS-CONSTRUCTION
A.
then
PERFECT
a
fully
from
I.i,
~ PH. intersection
of
it
is to
in
We
enquire
the
image
be
an
other
sion)
words,
an
P.
have
is
F,
= P.
be
On
theory
Ep2R
to
equality
holds
restrict
consideration,
hypotheses -
P
an
to
in
ensure
Epimorphism
of
free,
other
shall
an
(1.2).
Since suppose
that
(or,
epimorphism
correspond
to
has
perfect.
the
: we
seek
example
being
can
wish
~(PG)
in
context,
is
not
= P(~G), Exten-
Radicals.
R ~+ F -~
group
PP
group
%
extreme
Thus
free
we
We
Perfect
extension
conditions
which
that
Preserving
group no
what
to
epimorphism.
For the
under %G
only
So hand,
see
a
free
any
are
Ep2R,
presentation
of
non-trivial
although there
that
which
free
PF
=
i,
no
examples
surjection
let
a perfect
subgroups,
making
tPF from
with
and
=
i,
we
finite
finite
kernel
is
Ep2R.
Note
that
hypoabelian) In
of
among
G
n-th
forces
triviality
derived
groups
of
we
PG
must
(that
have
(PG)
is,
G
(n)
~ G (n)
is
fact,
The f o l l o w i n g
1.3
(i)
(iii)
The effect
2.
PG
=
1 and for
some
i
G (i)
PG
=
1 and for some
j
G(J)/Z(G
proof
that
is
an
G(i)/Z(G
Algebraic
a
three conditions are equivalent
:-
G is soluble;
(ii)
on
solubility
since
easy
(i))
exercise, finite
is finite;
save
implies
is finite.
(j))
for
a
G (i+l)
lemma
of
Schur
to
the
finite.
results
Our
main
group
G
epimorphism
purpose and
%
as
Proposition
of
these
to is
here
normal
the
is
We
maps
what
it
can
easy
to
that
if
}
then
~ must
also
be
Ep2R;
LEMMA
2.1
: G
be ÷
H,
so,
(in
: Ker
~
Several
(for
t:
about H ÷
under
Q
what
below) to
conditions Lemma
Since
said
(2.3)
sufficient
such
useful
question.
Ep2R
see
establish N
Ep2R.
find
following
Ep2R,
to
subgroup
: G --~ Q 2.3.
is
have
to the
relate
Again,
composite
that
an
col]ated one
composition
converse?
conditions
ensure are
3.7)
evidently the
conditions
~o9 will
it
: G ÷ Q ~ be
Ep2R
of is Ep2R as
well?
: Let ?
: G
+
H,t
: H
÷ Q
be epimorphisms
some finite n, (Ker
~)(n)
~
tPG.
such that~
for
Then
(a)
~ is
Ep2R; a n d
(b)
~ is
Ep2R i f
Proof.
Let
(2.2)
below,
lemma (Ker
~)(n)
j(n+l),
J ~ H
Thus
of
which
and
Ep2R
WPG
~PG
= PH
I am which
led
one
me
got
is
Ep2R,
By
PH
=
assumption
to
the
for
all
For
Prof.
K~
B.
immediate is
PROPOSITION
(ii)
the
other
map)
(iii)
- and
of
PH
t)(n)
(PQ). it
From is
is
(b),
the
known So
equal
= FQ.
when
that
j(n)
therefore
FQ (n)
and
~PG
9.
Then
=
last
Hartley
~o@
to
This is
the
both
~PG.(Ker
subgroup
have
image
PQ,
¢)(n)
lies
in
@PG,
leaving
be an
=
fact
that
P
whenever
application
2.3
: An
extension
argue
[j(m)K(n-l)
of
an
argument
where
P is a p e r f e c t
= j(m)K(n)
m,
(iii)
suggesting
m ~ n ~ O,
fixed
hand,
for
lemma.
J-~P
finite
each
from
the
(k)
-I
(1.2)).
namely
(@PG).Ker
this
following
: Let
j(n)
An
then
= ((tPG).Ker
to
= 0 results
identity
in
j(n),
implication
= P H (n)
2.2
Then
On
as
that
grateful
Proof.
j(m)
(after
lies
image
~
However
~ j(n+l)
perfect,
j(n)
n
~)(n).
(PH) (n+l)
thence
image
required.
LEMMA
group.
inverse
j(n+l)(Ker
same
Ep2R.
~o~ i s
the
deduces
~ H.
as
if
maps.
if
one
PH
because
=
the
Conversely, from
=
being
has
(a),
composite
PH
j(n),
PH
establishes
~
only
denote
j(n)
~ ~PG
whence
- PH.
and
induction
= p(m)
with
j(n-1)
Lemma
following
extension
on the
n.
The
latter
= j(m)K(n-l)
' j(m)K(n-l)
of
the
by
it
case
the
image
follows
of
that
] = j(m)K(n)
2.1(a)
(or
(b)
with
~
as
the
result.
N~
G --~ Q is Ep2R
it is split;
G (m) ~ N.PG for some f i n i t e m; N (n) =< PG f o r s o m e f i n i t e n; o r
provided
either
the homomorphism G ÷ Aut(N/PN),
(iv)
has
REMARKS. ful,
in
its
to
as
Finally,
example
the
study
Proof.
also
G
(i)
is
borrow
finite
so
Thus we
shall
weaker
of
§3
than
may
PROPOSITION
homomorphism
since also We
~ is
b)
[PG,N]
In
fact,
by
(2.4)
implies shall
case...
therefore
an
(1.2),
infinite reveals.
extension
since
For
PQ
in
fact
will
if
(ii)
~ Q(m)
the
we
for
course of
=
establish
be
proved
by
homomorphism commuting
Q
÷ Out(N/PN)
all
of
whiah
(iii).
PG/P. the
following
topological condition
is
diagram
Aut(N/PN)
extension and
by
then P(G/P)
÷
if
by
group
consequence
G ~ Aut(N)
~ An
The
N~
G @-~Q
only
if both
induces
the
trivial
PN. is
~ the
in
the
÷ Out(N)
Now
more
and
Ep2RI
it
PQ.
is perfect,
from
replaced
= PQ.
discussion,
shall
or,
use-
= ¢PG.
automorphism
seen
÷ Out(N/PN)
=
~
of
¢@PQ
immediate
converse
its
be
2.5
PQ
a)
whose
That
~
~PG
further
we
be
most
length.
a perfect induced
~ t(PG.N)
following
work,
(iv),
below. (iv)
that
: If P ~ G
extra
not of
~PG
the
derived
application
then
¢(o(m).N)
the
2.4
a little
fact
demands
of
may
automorphisms
¢,
perhaps
extensions,
finite
(iii)
a two-fold
the
is
perfect has
presentation
to
~
and
(ii),
= Aut(N/PN)/Inn(N/PN).
(iii)
kernel
outer
just
(iv) use
strengthening in
[4].
FQ
only
COROLLARY
means
in
(1.2)
that
make
By
in
the
is
central
a free
of
in Out(N/PN)
hypotheses,
whose
right-inverse from
m,
to
of
further
image
above
occurring
the
pursued
: Q ÷
the
extensions
ordinals
ones,
is
Of
application
generally, finite
hypoabelian
induced by conjugation,
is
only
necessary
Ep2R
equivalence demonstrate
if
and of
to only
[PG,N] this
prove if
this
N/PN~
= PN
and
notationally
result G/PN--~ [P(G/PN), simpler,
when Q
PN
is. N/PN] special
= I, (2.4) =
I.
LEMMA
trivial
t is Ep2R;
b)
[PG,N]
Z(N)
~
CH(N)
From giving next
%
fact
2.7
of A.
that
So
embeds of
centre
induced
the
extension
the
~ H/N is
in Out(N).
in Out(N) an
N~H~PQ
homomorphism
H/N.CH(N)
PQ
of
is Ep2R
that
there
whence
4(PCH(N))
Suppose
:
a)
[PA,B] PA
=
and
extension
This therefore
NnCH(N)
N has
derived
CH(N)
--~ PQ
is
length Ep2R.
at
that
PG
~ N.PCH(N),
=
most
i,
However,
too.
= PQ
(2.6)(h)
we
is
have
an
immediate
application
of
that
= PB,
(From
where
BC
B,
C are c o m m u t i n g
normal
sub-
= PC; and
and
(2.7)(b) we
: If
then
(2.7)(a)
may
(b)
instead identify
the extension
of
are
equivalent.
(2.7)(a)
H above
N ~
as
we
obtain
the
N.PG.)
G --~ Q induces
the
trivial
below,
for
PG = PN.PCN.PG(N).
the
lemma
we
P
: PA
Q
: [PA,B]
P0
is
verify
the
assertions
Pa'
Qa
a.
Evidently
gives
applying
2.9
prove
ordinal
[PA,C]
(2.6)(a)
map P Q + O u t ( N ) ,
for
=
: Conditions by
COROLLARY
To
A
PB.PC.
2.8
following.
P~-I
induces
i
Then
b)
Note
QB
=
lemma.
REMARK
of
: G -->> Q
the
observe
and
satisfied,
= P(N.PCH(N)).
LEMMA
each
so
image
the
to
we
and
the
Since is
that
[4],
H = N.CH(N)
--~ PQ.
the
PG
groups
precisely
so
restrict
in
N.CH(N)
(2.3)iii)
~ G,
we
as
Consequently
condition CH(N)
first,
kernel
H/N.CH(N)
trivial.
the
(a)
has
PN
if both
and
Proceeding
H ÷ Out(N)
with
N ¢-~ G t-~ Q
if and only
= i.
obtain
PQ.
makes
extension
PQ + O u t ( N )
a)
To over
: An
2.6
map
all that
~ B(a).C
true.
B < a clearly of PA
Pa,
since
=
[PA,PA]
.
~ B (a)
If
a
forces
~
is
a limit
that
[B(a-l).c,
of
ordinal, Qa
then
otherwise,
B(a-I).c]
the the
~ B(a).C
truth truth
.
of
6
On
the
other
hand,
because
PA
we
have
Q~
induction
=
implying
Since that
the
lemma
2.3
(and
the
all
[PA,B]
an
thereby
Qa"
=
of
interesting
us
check
PA
Now
the
the
have
(after,
proof
e.g.,
any
identify extension
these B~C=+
the
result.
easier.
So
transfinite
latter,
[PB,B]
~
require
[PA,B]
BnC
to
whose
be
central
proof
in
uses
A,
Proposition
A = B.C for normal
subgroups B,C of A.
length or Out(B~C)
or
= RB.PC.(B~C)
(2.10)
=
[PA.(BnC)
=
[PB.PC.(BnC),
=
[PB.PC,
of
of finite
does
indeed
imply
2.7).
For,
, PA.(BqC)] PB.PC.(B~C)] = PB.PC.
PB.PC]
(2.10).
From
~ B/BnC
the
decomposition
× C/BnC
,
(2.3)i))
perfect
= P(B/BnC)
radicals
D -->>D/B~C
PA.(BnC)/BnC whence
=
series
that
P(A/BnC)
We
even
,
then
A/BnC
we
PB
(2.7)
: Suppose
its derived
hypoabelian~
first
is the
generalization
PA.(B~C)
Let
~ B(a).C
2.7).
P R O P O S I T I O N 2.10
If B~C has either
[PA,B].C
From
B(a)
~ ~
Lemma
Out(B~C/P(B~C))
~
holds
converse
Pc'
hypotheses
has
Q
[PA,B.C]
Pa~
clinches
when
is
by
× P(C/BnC)
appealing
Ep2R.
z PB.(BnC)/BnC
In
×
to
(2.3)iii)~
particular,
PC.(BnC)/BnC
we
;
iv). deduce
Thus that
3.
Connections In
KIA
is
This
the
Whitehead
=
derived
classical
length,
fact
and
that and
has
been
ning
The
time) the
KIA
lies
and
[I
general
GLA
we
main
their
is
which
definition use
of
of
close
It
Here
led
the
to
two
of
ring
course
to
the
groups
of
the
plus-
Since in
Ep2R
A.
matrices.
a discussion
of
natural
examples
of to
was
Quillen
study
the
class
for
of
elementary
the
the
are
over
by
K.A = v (BGLA+). 1 1 plus-construction,
relation
definition
well-suited
radicals.
motivation
in
seen
the
group
generated
a member
have
= GLA/PGLA the
for
linear
subgroup
perfect
further
then topology
maps
(at
questions
the
concer-
(respectively
[i
(6.8)]).
Suppose
(3.1)
the
making
plus-construction.
(5.11)]
starting-point
the
which
thence
much
geometry.
present
on
EA,
preserving
construction there
as
[GLA,GLA],
epimorphisms
the
lemma
PGLA
EA
plus-construction
K-theory,
identifies
finite
the
algebraic
Moreover,
of
with
f
: X
+
has connected fibre.
Y
Then the commuting
diagram qx
X
qY
Y
is co-Cartesian
+
y+
If the fibre sequence
(3.2)
is Ep2R. F
÷
E
~
(with F,E,B
B
(that is~ F ÷ is also the homotopy
connected)
p+),
fibre of
is
then
is Ep2R. The
proof
whose
evident
This
suspicion
First, is
>
X
if and only if %l(f)
plus-constructive ~l(p)
>
needed
spaces)
trivial.
can
it
is
in
order
which
are
(3.3)
only if
of
~l(p)
(3.2)
in
[I]
irreversibility be
validated
possible to
in
to
two
the
doubt
argument on
the
precisely
what
fibre
Thus
further
sequences
[3]
reveals
condition (of
the
F ÷ E ~ B is plus-constructive
induced
converse.
ways.
those
plus-constructive.
Ep2R a n d
a diagram-chasing
considerable
state
characterise
A fibre sequence is
involves
casts
action
of
connected
following.
if and
P~I(E) on ~,(F +)
is
In with
the
~I(F)
special
G
sequence
both ~ is Ep2R This
comes
PROPOSITION
3.5
homomorphism
BN
÷
The group of
proof
BN +.
of
+)
Since
This be
some
base
groups
greatly.
with
is plus-constructive
PN
=
Then
i.
if and only if
i. to
implying
filled
by
the
the
as-yet
following
An extension N ~
: PQ
÷
(3.5)
of
(free)
PN
= I,
argument
unproved
part
corollary
to
of
the
main
with PN = I induces
G --~> Q
if and only if the fibre
Out(N)
(given
[3])
hometopy this
sense
has
is
dual
subgroup
simply
classes just
consists of
AUT(BN)
3.6
the
sequence
in
identifying
self-homotopy
the
equivalences
= wo(Aut(BN))
sects
the
ring A.
split By
with
matrix
over
finite
matrices,
A.
take OA
as So
G
A) A
which
is
indeed
above,
the
upper
: GLUT
~ is
fibre
sequence
base corollary, whose
are hypoabelian.
examples of
in
the
concerning condition
linear
group
triangular
÷ GL(A
certainly
abelian
and
following
Then
2
×
(2.5) on
GLUT
the
(after
2 - matrices
epimorphism
in
an
space
the
plus-constructive.
general
ring
multiplication N
every
is
important
matrices
making
total
is plus-constructive.
of
split
~
that
irredundancy
for
The
the
therefore
both N and O u t ( N )
the
epimorphism
(2.3)i) GLnUT
[2] group
provides
We
a
result
is
with fibre B N
K-theory
ring
that
There
Suppose
:
[PG,N]. on
groups.
the
demonstrates
a given
require
fundamental
it
[i p.27])
induces
to
sequence
Algebraic (3.4);
not
spaces.
hypoabelian
COROLLARY
over
does
classifying
every fibre
and
of
[5 p.42].
space in
BQ
spaces
is plus-constructive.
BQ
AUT(BN
Out(N)
=
is
classifying
[3].
trivial BG
+
close
gap
are
be a group extension,
BG
[PG,N]
F,E,B
simplifies
-~>Q ÷
very
The
of
BN
and
above.
÷
this
Let N~
(3.4)
theorem
where
hypoabelian,
the fibre
(2.5)
case
N
@
A)
Ep2R. where
corresponds
subgroup.
of
general
Now M
is
N an
to However,
linear
= Ker
~
inter-
arbitrary
addition N
fails
n
of to
x
n
-
commute
with
PGLUT
= EUT.
For
example, 0
0
1
0
in
GL2UT
the
matrices
12 NnGL2UT 12 1 ~ el2 image
and the
sequence
12 of BN
allows @
E(A ÷
the
BGL(A
•
(since
sequences
[i
BN +
(12.3)],
The
[6].
algebraic
÷ EA
is
has
trivial
isomorphism.
of
exact
long
The in
second
of
have
Fp+
= F + , one
on
fundamental
homology these
the
(with two
a lemma
based
LEMMA
To exact
prove only
to
map
arise
= Fp
distinct
is
fulfilled
induced
$
map
is
difficult the
to
Fp+
B
and
an
the
relative
is
aspects in
(3.2)
in
terms
of
(3.8),
of
Recall
it
and thus
above
terms
Fp+.
that
P~I(F)) It
as
[I on
However, if
so
A-
K.A I.
F ÷
coefficients).
the
structure
A 1 the
thereby
F ÷ E ~
kernel
signi-
KA0(BG)
perfect).
a converse
F
÷
induce
an
consists the that,
an
to
epimor-
isomorphism
to
be
the
original
(3.9)
on
expected
below.
that fibre First,
(2.1).
: E ÷
(3.7), in
the
from
further
makes
and
sequences
(with
B
A ~ B ~
respect
we
is a fibre
C
(with connected
Then ~l(p)
÷ A.
K.A
fibres
of
groups
: Suppose
3.7
and that a map p : A×BE
fibre
map
A--~
which
BEA~,
+
whose
K-group
all
fibre fact
Mayer-Vietoris
natural
define
the
BGLUT
plus-constructive
concerning
expectation on
fact
linking
requires
should
This
the
the
after
only
BEA + ÷
(homotopy)
arbitrary
data
sequence.
is of
is
(3.4)
:
K-theory
algebraic to
is
(2.5))
This
Bt +
epimorphism
(EA B*
discussion
between
is
this fibre
of
one
from
contractible),
lower
the
ring
centre,
a characterisation
phism
any
not
of
Ep2R
sequence
relation
p'
12,13].
chs
and
(by
plus-constructive.
equivalence
G to
[i
for
the
not
additivity
enables
Hence
non-trivial, is
patently
latter
Again, of
is
a group
evidently
description the
the
generally, EA 1
A)
homotopy
= BN
of
K-theory
More
is
@
the
commutator.
non-extension and
ring
Out(N)
BGL(A
the
ch.3]
non-trivial
in
of
representation
since
A)
BGLUTB-~t
embraces
: EA
have
existence
A) +
ficance
~ E2UT
is Ep2R consider of
the
sequence
homotopy
fibre)
if and only if ~l(p the
first
following pairs
of
commuting horizontal
')
with F~I(C)
=
I,
induces is. diagram, maps.
which
10 ~l(f') ~2(C) + ~I(A×BE) -->>
Im vl(f')
I ~I(p')
~l(gop) -+
~ ~I(E)
I ~I(P)I
I ~I(p)
~l(f) ~2(C)
Since
~2(C)
(b)
that
(a)
occurs
is ,
wl(p
abelian, ) is
that
Ep2R
if
EP
Ker
Lemma 2
~l(g)
2.1
yields
R precisely
when
PwI(C) and
~l(g)
-~
precisely
hypothesis wl(p)l
÷ ~I(A)
the
when
if
PROPOSITION 3.8 :
~+
~I(B)
(a)
that
wl(p)
i
~l(POf
restriction
= 1 forces
only
to
Proof. is
affected
of
course
Then
B+
back
over
By
~l(p)
by P~I(B
is
[2 Lemma
2.1]
pulling-back +)
= i).
contractible,
) = ~l(f~p') is
lie
B + B + to
and
over
induce
is,
EPZR.
inside
and
which
However,
Ker
wl(g),
by the
making
sequence F ÷ E ~ B with F conn-
the the
lemma
the
a map
of
above,
fibre
neither
sequence
assume
that
F ~
~I(F) ~ ~l(Fp+)
is
Ep2R.
is
Therefore so
is Ep2R,
is.
For any fibre
if and only if
~I(C)
~l(f')
wl(p)l
FWl(B)
~+
ected, the induced f u n d a m e n t a l group homomorphism surjective
~l(C)
B
fibre
is
AB
fibre
÷ B + B+
already
sequence
an
over
(since
acyclic
÷ E+ ÷
Fp+
sequences
assertion
space.
B + pulls
B:
Fp+ : E +
\
/
E g E+xB
p\/ B As fibre only
the
maps
[~ p.35],
of it
total
follows
spaces that
and
of
Wl(Fg)
fibres
maps
share
~I(F)
a common
onto
Wl(Fp+)
homotopy if
and
if
~l(g) : ~I(E) + ~I(E+xB) : ~I(B) x ~I(E)/P~I(E) is
surjective.
so
that
~I(B)
Now ~ w/K.
write There
~ for is
Wl(E), a map
P of
for group
P~I(E)
and
extensions
K for
Ker
~l(p),
The metric (y~6)
=
( ~ 6 ) -I
manifold cules
by
We
(gij)
shall
the
use
a more
frames,
= {(x,y)
and
latin ones
to the
while
calculus,like
at other
covariant
x . occasions
we
derivatives,
or-
etc.,
w h a t e v e r is m o s t c o n v e n i e n t . o d e n o t e b y U its i n t e r i o r a n d
we
X , we
in the p l a n e ,
denote
a geodesic
ball with
: d(p,q)
d is the d i s t a n c e and
-< R}
notation
minology
p 6 X a n d ra-
function
,
o n X.
l o w e r b o u n d s for the s e c t i o n a l 2 2 denoted by K and -e , i.e.
__ O
for some for w h i c h
the D i r i c h l e t
the u n , w e
find a s e q u e n c e
, x n + xo E Z I ' r n ÷ 0
Pn £ ~2
' Pn ÷
UnlB(Xn,rn)
problem
can
p 6 Z 2 , en + 0
is n o t h o m o t o p i c
(4.4.4).
, with
to
,
45
In c a s e I), w e r e p l a c e let problem (4.4.4) for and , u s i n g solution
of
u n on B ( X l , r n , I) by x = xI
and
the i n t e r i o r m o d u l u s (4.4.4)
(cf. Thm.
< ~ . By Lemma
(4.4.5)
r = rn, I
.We can a s s u m e
of c o n t i n u i t y
4.1)
1 d e n o t e d by u n , c o n v e r g e
maps,
the s o l u t i o n of the D i r i c h -
estimates
rn, I + r I
for the
t h a t the r e p l a c e d
uniformly
on B ( x 1 , ~ - ~ ) ,
for any 0
y
= y~BD2f(Ux~,UxS)
as w a s n o t e d
Proposition
f)ou
coordinates
a consequence,
property,
~(grad
frame
A(fou)
a harmonic,
+
if u is h a r m o n i c ,
A(fou)
o r in l o c a l
~) e
e
J~ger
then
e,u
e ~ is an o r t h o n o r m a l
In p a r t i c u l a r ,
5,2.
an e l e m e n t a r y
sequel.
e
lows
functions
is v a l i d .
(5.1.1)
As
convex
the
and Kaul following
problem
uniqueness
for h a r m o n i c
u. : ~ ÷ N a r e h a r m o n i c l N C 2(~,N) , ~ is a b o u n d e d d o m a i n a n d ui(~)
theorem
maps.
maps
of c l a s s
c B(p,M),
where
of
55
B(p,M) with
is a g e o d e s i c
radius
ball
M < z/2K
in N
, disjoint
(K 2 is an u p p e r
to the c u t
bound
for the
locus
of p a n d
sectional
curvature
of B(p,M)). Then
the
function
@ , q
=
0
,
then s ' (p)
~5.2.3) (Here,
(x,x,>
we s -s
K
- (p)
(Ix(o) 12 ÷ Ix(p) ] 2)
0
_ O
,
- s"
Ixi
= s(
(Ix121x' 12
)
+ O
g is n o t d e c r e a s i n g
IXl '(p)
= lim ~+O
on
[O,p]
IXl '(p-e),
.
, and d e f i n i n g
IXl ' (0) = lim s+O
IXl '(s)
,
we c o n c l u d e
O < g(p)
- g(O)
= s(p) IXl ' (P) - s' (p)IX(p) I-
- s(0) Ixl'(o) -- < x , x ' >
+ s'(0) Ix(0)l:
(~)-
<x,x'>
(o)-
s'< (p) s ei(Y) nor tan v. : = v. - v. 1
Then,
1
since
(i = 1,2)
1
P > O ,
grad d(y)
= e I (y)~e2(Y)
grad QK(y)
,
= SK(P) (el(Y) ~ e 2 ( Y ) )
D2QK(y) (v,v)
,
and
= < D v grad Q < , v >
!
(5.2.9) If Ys(t)
= SK(P)<e1(Y)~e2(Y),Vl~V2> is the g e o d e s i c
2 + s
-
)dt
O
(note that there X,7'>
is no b o u n d a r y
= 0
=
since
)
Since X s a t i s f i e s D2d(v,v)
term,
(5.2.6),
~(Ix'l2
we can apply
Lemma
5.1
to o b t a i n
+ <x,x">)dt
O
'
p
=<x,x > io
>
!
s sK (~) and thus with (5.2.12)
+
nor 2 v2 )
2
nor
I'lv2
(5.29) D2Q S'K (p) ( < e I ~ e 2 , v l e v 2 > 2 + Iv nor I I2
21v°rlIv°rl If v = O ~ u
nor
s ip) lVl
, (5.2.12)
D2QK (v,v)
implies
> SK(p) < e 2 ( Y ) , u
>2
! + S S'K (p) < e 1 @ ~ e 2 ' v l ( ~ V 2 >
(I -s ~(11
+ s~K(p))
>2 _ (1 -sK(p))'
<e1~e2,vlev2
(Iv112 Iv212) +
2QK(y) 1
i~rad ~[ 24
-
2
on ~ .
since
= I _ < 2 Q < ( p , u i(x)) ,
we o b t a i n (5.2.16) Finally,
A4 i(x) by
=< - < 2 4 i l d u i 1 2
(5.2.13), 2
(5.2.17)
_1 Igrad ~I 2 ~2
> -
grad
Igrad
4 I12
Igrad
+
4212
+ 2 41
to2 2
I log 0 , ~ grad
grad 42
grad 41 log 0 + 41
+
42
>
Putting k(x):
I = ~ grad
grad 41 log 0 +
grad 42 +
41 and p l u g g i n g
(5.2.15),
(5.2.16),
42 and
(5.2.17)
tain A log @ +
~grad
log @ , k ( x ) >
> O
into
(5.2.14),
we ob-
62
Therefore
log
Thm.
5.1
tion
that
8 is a s u b s o l u t i o n
follows
from'E.
8 has
a positive
of
Hopf's
a linear
maximum
maximum
elliptic
principle,
in t h e
interior
problem,
if t h e
equation,
and
since
the
assump-
leads
to a c o n t r a -
diction.
5.3.
Uniqueness tive
In this
and
section,
5.1
we
the
which
Yau
in
5.2:
show
that,
assumption [Hm].
shall
calculations
and
Theorem
skip
to
to Hamilton
[Sy3],
(those
Schoen
Dirichlet
we want
can
is d u e
Yau
the
We
enable
in t h e
if N h a s ui(~)
shall
has
us
nonposi-
sectional
in T h m .
an observation
to carry
context
nonpositive
c B(p,M)
use
over
of Thm.
the
proof
5.2 were
5.1. of
also
This
Schoen
of
Thm.
given
by
[SY3]).
Suppose
N has
nonpositive
sectional
u.: ~ ÷ N , i = 1,2 , a r e h a r m o n i c m a p s , l u i 6 C2(~,N) n C°(~,N). If u 1 1 ~ = u21~ topic,
image
curvature
curvature, result
for
~
curvature,
is a b o u n d e d
, and
u I and
and
domain,
u 2 are
and
homo-
then
uI ~ u2 . Proof: N has
Let
denotes
the
isometrics
thus
=
N be
the
sectional
distance
on N
~(x,y)
and a
~ and
nonpositive
universal
covers
curvature,
function
of N
~2 x N
is
of
~ a n d N,
smooth
. ~(N)
on N
acts
resp. x N
Since , where
as a g r o u p
of
x N via
(~(x),~(y))
induces
for
~
6 ~i (N)
,
a function
"a': N x ~/~r 1 (N) ->- ~ . Let
F: ~
u I (x)
÷ N and 7
x
and
[0,1 ] ÷ N b e F(x,1)
thus
obtain
6 ~I (~) ' t h e r e
(5.3.1)
for
all
6 ~
.
for
liftings
exists
Ul (Y(x))
x
a homotopy
= u2(x)
~
x Ul
6 ~
between . We
= F(-,O)
u I and
choose and
u2
u 2 , with
a lifting = F(-,I).
6 71 (N) w i t h
= ~Ul (x)
and
u2 (Y(x))
= ~u2(x)
F:
F(x,O) ~ ×
Then
=
[0,1 ] for
any
63
~: ~ ÷ N x N , d e f i n e d
Thus by
(5.3.1)
induces
b y ~(x)
a harmonic
U : f~ ÷ N x N I z I (N)
=
(~1(x),~2(x))
, is h a r m o n i c
and
map
.
Then
O : = 1/2d2ou
is a s m o o t h proof
function
of Thm.
(5.3.2)
5.1
sup
on ~
, and we can
to s h o w t h a t
8 ~ sup
carry
e satisfies
over
the
arguments
the m a x i m u m
of t h e
principle
@
q.e.d.
5.4.
Uniqueness sitive
The
arguments
uniqueness
Theorem
5.3:
a)
Suppose
homotopic
the
curve
, t £
maps
[0,1]
ht(x),
t 6
We construct
5.2,
From
(5.2.15)
TxM
assuming
T
we
sectional
N
. From we
s~ive, (i.e.
X'
a function
infer
for
has
nonpo-
see
following
[SY3].
without
to u u n l e s s
that
curvature,
, then
maps
and with
8 in t h e
u I and u 2 are again
boundary r
u(M)
Then
is c o n t a i n e d
that
8(x)
exists
therefore
between
that D2d(v,v) since
the Jacobi
the
a smooth
length
arc,
= 0
s a m e w a y as
Yl a n d Y2
field X defined
in
of
is a
where
' then we
v = v1~v
curvature
.
maps.
by the maximum
v i = d~i(ee),
, where
of x
in the p r o o f
harmonic
constant
sectional
fa-
~arame-
independant
= I/2 d 2 ( u 1 ( x ) , u 2 ( x ) )
yi = ~ i ( x ) ,
and moreover
and u I and u 2 are
h O = u I and h I = u 2 , and
homotopic
i = 1,2
and
there
with
on M and
5.2,
(5.2.11)
= O)
the in
is for a n y x £ M a 9 e o d e s i c
, a n d y = Y x is the g e o d e s i c of Thm.
shown
manifold
homotopic
to a r c l e n g t h
that
function
If w e p u t
the proof
image
a n d u: M ÷ N is h a r m o n i c .
f r o m M to N
[0,1]
Proof:
subharmonic
map
, of h a r m o n i c
proportionally
Thm.
also yield
as w a s
M is a c o m p a c t
ha~ nonpositive
trized
ciple.
if the
.
harmonic
ht
[Ht],
curvature,
harmonic
of N
If N o n l y
mily
solutions,
section
of H a r t m a n
sectional
is n o o t h e r
in a g e o d e s i c b)
for c l o s e d
of the p r e c e d i n g
theorem
N has negative there
results
curvature
e see
prin£ from
2 6 TyIN
of N is n o n p o -
(5 2.10)
is p a r a l l e l
64
"'<X,R(X,~')~' 2 :
(5.4.1)
Since
G is c o n s t a n t ,
0
d ( ~ 1 ( x ) , ~ 2 ( x ) ) =:
p is i n d e p e n d a n t
of x
, and we
define
ht(x)
= Yx(~)
t 6
[O,I ] and x
6
J
i
Since
X
= O and
(5.4.2)
where
dht(x) ( e )
Yx
= d~2(e
).
~I(N)
exists
htoo
and
e
Yx
by
6 ~I(N)
M ÷ N
From
Yx(O)
since
on N x N
= d ~ 1 ( e ~)
and Yx(p)
X is p a r a l l e l ) .
, we
see
that
for
o
£ z1(M),
with
[O,1 ]).
induced
maps
,
(5.4.2)
and
vector
e(h t) (x) = I/2 TxM)
since
Yx is p a r a l l e l ,
is i n d e p e n d a n t
of t
, and
In p a r t i c u l a r ,
any
the h t have
would with
all
exist
Sampson
proved.
For
sectional Since
a),
we o n l y
curvature,
Yx
independant
infer
that
maps
h
, homotopic than
have
Namely,
dht(x) (e ~)
is a
Thus,
to note
that
if N has
implies
that
X and
, and
t
>
energy.
the
same
energy.
otherwise,
b)
u 2 are h o m o t o p i c arc
also
to u I ,
theorem
of E e l l s
is c o m p l e t e l y
strictly
= O by c o n s t r u c t i o n ,
in the g e o d e s i c
there
negative
Yx! are p r o p o r t i o n a l .
to Yx' ' for any e
if u I and
same
to h t and thus
have
<X,Tx
the
h t by the e x i s t e n c e
is a c o n t r a d i c t i o n .
is p r o p o r t i o n a l of x
harmonic
h t have
(5.4.1)
hand
to be c o n t a i n e d
the m a p s
to be h a r m o n i c .
energy
which
on the o t h e r
therefore
map
smaller ~S],
all
two h o m o t o p i c
a harmonic
strictly
we
f i e l d a l o n g the g e o d e s i c f r o m ul (x) to u2(x). T h u s n [ Idht(x) ( e ) I 2 (where e is an o r t h o n o r m a l b a s e of ~=I
Therefore,
have
Yx w i t h
hO = u I , h I = u 2
parallel
and
along
is p a r a l l e l ,
(t £
we o b t a i n
,
isometries
= ~oh t
Therefore,
ht:
field
(Note t h a t acts
holds,
= Yx(t)
is the J a c o b i
Because there
(5.4.1)
6 TxM
X = O
, and
. Thus,
harmonic
Yx
maps,
is they
y = Yx q.e.d.
65
Remark:
In the t w o d i m e n s i o n a l ,
o r e m of E e l l s - S a m p s o n
5.5.
Uniqueness
which
we can replace
w e do n o t p r o v e
and nonuniqueness
the a p p e a l
in t h i s
for h a r m o n i c
maps
to t h e
the-
b o o k b y Thm.
4.2.
between
closed
surfaces We now
look more
between
closed
carefully
surfaces,
t h e n u is n e c e s s a r i l y (unless
it is c o n s t a n t , the o n l y
12.1.
a family
case
Therefore
ZI ÷ Z2
a conformal
quently Cor.
at u n i q u e n e s s
u:
which
maps,
is the
or a n t i c o n f o r m a l
of h a r m o n i c two -sphere branched
maps S2 ,
cover
is t h e
c a s e if X ( Z 2) ~ O , a n d c o n s e is Z2 = S 2 ), as w e s h a l l see in
of i n t e r e s t in e v e r y
of harmonic
properties
" If ZI
homotopy
which
class
depends
on
e 6
[$2,$2],
we obtain
(2 + 4 d e g e) p a r a -
meters. If ~I or ZI o r
Z2 is a f l a t
Z2 a g a i n
O n the o t h e r tries
gives
hand,
homotopic
formal
with
torus
a surface
to the
respect
with
ter
I , nonuniqueness
We
the
also observe
also
isometries
domain
cannot
homotopy
(or c o n f o r m a l
cause
class
any
A-priori
6.1.
Composition
In t h e c a s e ,
Lemma
situation,
of nonzero
6.
position
Proof:
of
be c a u s e d Theorem sense
by
that
if the g e n u s
if Z2 has isometries 12.1),
its d e g r e e
cf.
of h a r m o n i c
genus
to
grea-
of the
image
any map
from
is zero.
Lea%Ma 6.1)
maps
be con-
and had
of Z2 is g r e a t e r
automorphisms,
isome-
~uld
structure
(cf. Thm.
any
t h a n one, of
the
in a p r e s c r i b e d
degree
between where
surfaces
of g e n u s
the h a r m o n i c
map
at least
two,
in a g i v e n h o m o -
is n o t u n i q u e .
C1'~-estimates
where
of harmonic
maps
X is a s u r f a c e ,
with
conformal
we have
the
maps
following
useful
com-
property:
6.1:
Suppose
M ÷ X is a c o n f o r m a l again
not have
s u c h an i s o m e t r y conformal
g > I . Thus,
nonuniqueness
in the c a s e o f m a p s
do n o t k n o w
topy
in t h e
case,
an i s o m e t r y
class.
Actually, we
since
by K n e s e r ' s
in t h e n o n t r i v i a l
u with
g > I does
since
cannot
S 2 o r T 2 t o Z2 is t r i v i a l Thus,
of g e n u s
underlying
identity,
that
composing
map.
identity,
to the
coincide than
T 2 , then
a harmonic
harmonic,
u
6 C2(X,Y)
map
between
a n d E(u°k)
We only have
is h a r m o n i c the
and dim X = 2
surfaces
M and X
= E(u)
to n o t e
that
(1.3.1),
namely
If k:
, then uok
is
66
u i) I /Y
~ ~x ~
remains valent
(~yy y~B
valid, one,
(1.3.4)),
if we
say
and
replace
l(x)yes(x)
the
same
the m e t r i c
, l(x)
is true
> O
%~ B(x)
by a c o n f o r m a l l y
, in case
dim X = 2
equi-
(cf.
also
for
~u i ~u j
I
E(u)
+ ya~F i ~ uj ~ uk = O jk ~x ~ -~x 8
~x ~
= ~ ~ ¥~Bgij
/~ dx I dx 2 ~x ~
q.e.d.
Cor.
We
6.1:
On
shall
prove
later
on e x p l o i t
a -priori
for the derive
case
a -priori
taking
6.1
conformal
in the
for h a r m o n i c
maps
maps
between
the d o m a i n
is the u n i t
estimates
from below
for
maps,
us to a p p l y
the e x i s t e n c e
i.e.
these
theorem
the
maps
We
shall
surfaces,
disc.
first
T h e n we
conformal
estimates
for c o n f o r m a l
way.
functional
in p a r t i c u l a r
special
are h a r m o n i c .
following
where
harmonic
enable
domains,
Lemma
estimates
special
of u n i v a l e n t will
twodimensional
shall
determinant maps,
which
to the g e n e r a l
case,
of c h a p t e r
3 into
account.
6.2.
A maximum
The p u r p o s e mates not
of the p r e s e n t
for g e n e r a l
only
could
appeal
harmonic
to Thm.
however.
dary values
map
5.1
mates
obtained
trary
twodimensional an e x i s t e n c e
Schauder
degree
following
dimension. sional general
case
4.1
chapter
domains
estimates
all
for
4.1,
simplicity
for
that
the d o m a i n
for
be
stated
image and
of e x p o s i t i o n ,
we
conditions
that
goal
in
the b o u n -
the C 1 ' ~ - e s t i -
D and extended
9 will
this
these
a different
energy,
eventually
assumption
of v a r i a t i o n a l
only minor
under
ball,
controlled
Of c o u r s e ,
to r e q u i r e
finite
hold
4.1.
esti-
in a c o n v e x
we a l r e a d y
We h a v e
we h a d
with
without
instead
requires
of Thm.
to s h o w
a -priori
contained
for w h i c h
in c h a p t e r
theorem
of t h e m w i l l
is to o b t a i n
image
minimizing.
theory
Some
ones,
and
in T h e o r e m
to p r o v e
The
ones,
an e x t e n s i o n
in this
now
with
in the p r o o f
is e n e r g y
While
possess
maps
minimizing
of c o n t i n u i t y
every
chapter
harmonic
for e n e r g y
the m o d u l u s
mind,
principle
to a r b i -
enable
using
us
Leray -
methods.
manifolds proved
although
modifications.
of a r b i t r a r y
only
They
for t w o d i m e n -
the p r o o f also
hold
in the for
87
domains cf. We
e.g.
arbitrary
with
Jost
Lemma
the
[J1],
6.2:
, e.g.
B(p,M)
If
is
Suppose
~ is is
max x£~ is
Proof:
which
For
by
c in V w i t h
the
proofs
are
different,
principle
which
can
be
for
in B ( p , M )
~
p,q
(cf.
6 V
q
cut
locus
6 B(p,M)
d(u(x),q)
M
boundary
< z/2
O and R > O,R ~ R
( d e p e n d i n g on c , R , M
d 2 ( g ( x ) , P t ) on ~D N ~D(Xo,Ro)
(6.5.6)
Wt,R(y)
for all y 6 D(xo,r). found,
values
e.g.,
in
o
by a s s u m p t i o n .
, there exists
, and the m o d u l u s w i t h the p r o p e r t y
~D n
some n u m b e r
of c o n t i n u i t y
r = of
that
=< W t , R ( X o) + e This
is a r e s u l t
[GT], Thm.
If d 2 ( g ( x ) , P t ) is H ~ i d e r
from potential
8.27).
continuous,
w e even h a v e
theory
(and can be
71
(6.5.7)
where
Wt,R(Y)
e
, ~ depend
We n o w w a n t
we
e
to a p p l y
~:
=
~
e:
= M2((I-
with
for
~ is the
some
radius
where
r is the s a m e
Then,
with
=
by L e m m a
R
smallest o
=
O
harmonic
map,
where
8~
Therefore,
there
(7.2.11)
exists
IJ(x°)I
Putting
p: = m a x
(7.2.12)
x O 6 B(O,o)
with
_-> ~/~
(r,o),
IVu(x)l
we h a v e
< c
for x 6 B ( O , 1 / 2 ( 1 + Q ) )
by Thm.
6.1
,
c = c(M,~,-~/~ replace
desired
hand
by
(7.2.11)
, c 2 by ~/z
on the
left hand
side of
(7.2.13)
to o b t a i n
estimate. q.e.d.
Corollary
7. I:
with
u(D)
c B(p,M),
Suppose
r 6
(0,1)
B(p,M)
that being
u: D ÷ Z is an i n j e c t i v e a disc with
M < ~/2
6 -I =
where
6
= ~(M,~, E}
,
section,
f o r the
functional
Theorem
7.2:
B(p,M)
Suppose
we have
we want
determinant
of
the b o u n d a r y [JKI].
u: D ÷ E is h a r m o n i c ,
is
~u(D)
= u(~D)
a disc with and
radius
t h a t g:
M
a n d u(D)
< ~/2
_-~ ,
estimates
In t h i s
where
d(q,~u(D))
(7.2.14)
meas
7.3.
u is u n i v a l e n t
dg(~)
d~
t h a t g(~D)
following
f o r all ~0 6 ~D .
is s t r i c t l y
estimates
for
convex
the @ e o d e s i c
w.r.t,
u(D),
curvature
of
and
that
g(~D)
83
(7.3.2)
O < a I < ~i I
~I = ~ 1 ( ~ ' K ' M ' T ' a 1 ' a 2 ' b ' I g l c 1 , e )
Proof:
We d e f i n e h(q):
(7.3.1)
and
(7.3.4)
(7.3.2)
A(hou)
This will enable h ou at b o u n d a r y Hopf.
for all x 6 ~D ,
This
(T is g i v e n
= -d(q,3u(D))
imply
for q
6
~u(D)
in Thm.
6.2).
for q 6 u(D). (cf.
(2.5.1),
(2.5.2),
(5.1.2))
=> alb2
us to g e t a l o w e r b o u n d p o i n t s w i t h the a r g u m e n t
assertion
in turn i m p l i e s
for the r a d i a l of the b o u n d a r y
(7.3.3),
taking
derivative
of
l e m m a of E.
(7.3.1)
into
account. The constants
a 2 and < c o n t r o l
then d e t e r m i n e
how
a n d free o f d o u b l e p o i n t s . fore
find a neighborhood
C 2 function with Suppose
x
o
Using now
x ° 6 3B(Xl,rl),
Defining
Taking
V ° of
strictly
Cor.
a n d Cor.
6.4, w e can c h o o s e
2 rl = ~ - - a152(1
function
for x 6 B ( X l , r I) y(x)
(x_x I )2 2 ) rI
-
f
we have
= -I/2 alb2
and consequently
by
A ( h o u + y)(x) Moreover,
we can t h e r e that h is a
some d i s c B ( X l , r 1) c D ,
in s u c h a w a y t h a t
the a u x i l i a r y
A¥(x)
the p r o p e r t y
convex
level c u r v e s on U(Vo).
A(hou) (x) _~ 1/2 alb2
~(x):
and a I and strictly
6.2 into a c c o u n t ,
~D in D w i t h
convex
of u(~D),
curves of h r e m a i n
6 ~D . (7.3.4)
(7.3.5)
focal p o i n t s
long the level
,
(7.3.5)
> O
on B ( X l , r l ) .
via
84
(hou) (Xo) + Y(Xo)
= O
and
(hou) (x) + ¥(x)
=< O
on
~B(x
,rl), I
since
by a s s u m p t i o n
sumes
nonpositive
The m a x i m u m
principle
the d i r e c t i o n
and
of
the
(hou + y ) ( x
~r
u is m a p p e d
values,
o
onto
the
side
a n d y l ~ B ( X l , r I) = O
now
controls
outer
normal,
) > O =
the
of
~u(D),
where
h as-
e
derivative
of h o u + 7 at x
o
in
namely
,
thus
(7.3.6) (7.3.6)
~--~ and
(hou)
(7.3.1
(7.3.7)
) > rl alb2
(x o
by
-~-
definition
of y
.
imply rI ~ - alb3
IJ(x o)
=:
~i 1
q.e.d.
Corollary where
u(D)
Suppose Then
7.2:
c B(p,M),
that
for all
(7.3.8)
Assume
g:
a n d B(p,M)
= ul~D
6 C 1'e
is a d i s c w i t h , and
that
harmonic
radius
7.3.1)
and
mapr
M < z/2
~
,
~2 = 62(~'
is d e f i n e d
6 3 can be
< M
< 2c14
IJ(u) l c14
d(u(x),p)
chosen
o
in
1/2631 } in Thm.
9.2,
uniformly
and
6 3 in Thm.
for the
family
9.3
(note
of b o u n d a r y
t h a t c14 values
~(.,~)). Lemma
9.4:
The
into itself
transformation
is of L e r a y - S c h a u d e r
~ ÷ ~(u,l)
= ~ - H (5,1J
of
C I (~, 2)
type.
Moreover,
(9.4.6)
deg
(~(~,l)Yo,O)
for all
= I
I 6 [0,1]
In p a r t i c u l a r ,
(9.4.7)
has
~(~,I)
a solution
Lemma
9.4
Proof
of L e m m a
type,
follows
if ~(u,l) Y
o
in Y
implies
We now
c Yo
lution
o Thm.
9.4:
show
That
that
' and o
~ 6 To
, and
~ 6 Y
Consequently,
(~(~,l),Yo,O)
~ 6 K(1)
thus
9.3.
the t r a n s f o r m a t i o n
9.1.
deg
= 0 for s o m e
~ 6 ~Y
9.5 v i a L e m m a
from L e m m a
. Consequently,
K(1)
= O
o
, then
by L e m m a
is of L e r a y - S c h a u d e r degree
is i n d e p e n d a n t
is w e l l of I
definedh Indeed,
]J(u) I > 0 in ~ by
definition
9.3.
9.3 i m p l y
. Therefore,
the d e g r e e
the
Thms. ~(u,l)
is i n d e p e n d a n t
9.2 a n d
= O cannot of ~
.
have
of
a so-
98
It o n l y
remains
(9.4.8)
deg
If we
define
~(~,0)
where
the
By Lemma
with
(~(u,O),Yo,O)
= 1
, then
= ~(u, 1),
transformation
¢ was
defined
in 9.3.
therefore
(~(U,O) ,Y,O)
= I .
that Y
c Y (Y w a s d e f i n e d in 9.3, too). o h a n d , b y the u n i q u e n e s s t h e o r e m of J ~ g e r - K a u l
On the other 5.1),
that
g(~0) = k(%0) for R0 6 ~
9.2
deg
We note
to s h o w
any
solution
the one
~
6 Y of the e q u a t i o n
solution
Consequently,
any
in Y . Hence o o f the L e r a y -
Schauder
we know
solution
(9.4.8)
already,
~(~,O)
namely
in Y o f ~ ( ~ , O )
follows degree
9.2
(cf.
p.
(of. Thm.
to c o i n c i d e
the c o n f o r m a l
= 0 is a c t u a l l y
from Lemma [D1],
= 0 has
and
map k
.
contained
the e x c i s i o n
property
67).
q.e.d.
Remark:
Actually,
preceding values,
10.
proof
cf.
by
one
using
can
dispense
a more
of the c o n f o r m a l
geometric
variation
m a p k in t h e
o f the b o u n d a r y
[J1].
Harmonic
coordinates.
C 2' ~ - a - p r i o r i
estimates
for harmonic
maps 10. I. E x i s t e n c e In t h i s
chapter,
coordinates functions. optimal bounds
we want
on s u r f a c e s , It w i l l
regularity
coordinates.
to p r o v e i.e.
turn out
Riemannian
bounds.
properties.
normal
involve
Actually,
in
This
coordinates,
metric
with
is o n l y H 6 1 d e r
continuous
these
and regularity
with
be
displayed
(cf.
one
with
coordinate possess
can o b t a i n ±n t e r m s the
Ca -
only of
fact that
f o r the
for
Christoffel
[KI]).
the
continuous
in n o r m a l
of harmonic
coordinates
symbols,
contrasted
even L-bounds
HSlder
harmonic
harmonic
In p a r t i c u l a r ,
derivatives
[JK1 ], t h e r e w a s
twodimensional
existence
Christoffel
should
curvature
C I 'a - e s t i m a t e s
coordinates
that
f o r the c o r r e s p o n d i n g
curvature
symbols
of harmonic
following
example
curvature,
coordinates,
which
of
a
itself
but not better:
99
ds 2 = d r 2 + G 2 ( r , ~ ) d ~ 2
w i th
G2(r'~)
F o r this
{~
2( I + r 2 s i n ~ ) 2
=
forO
Grr
F'| -6r~ine~_e = ~ I + si,L ~
G
reason
ordinates
[JKI]
f o r O =< ~0 _~
does
the
not involve
detailed and
a disadvantage
the s i z e
increases.
is d i f f e r e n t
the e x i s t e n c e
Theorem
theorem
10.1:
being
of
class
C 1'e
Ah = 0
noted
by
in
coordinates
of these
Suppose
. Then
respect
c a n be
to ~
balls
in t w o
.
f o u n d in regularity
de T u r c k - K a z d a n
[JKI].
In h i g h e r
is t h a t
decreasing
dimensions,
co-
dimen-
(at l e a s t w i t h only
as the d i m e n s i o n
since here we
can
use
8.1 t o o b t a i n
the t o p o l o g i c a l
h: B ÷ D , D b e i n g
of G with
a v a i l a b l e ) Dne can p r o v e t h e i r e x i s t e n c e
on s m a l l b a l l s , This
for K in n o r m a l
t h a t the o p t i m a l
were
explicit
of h a r m o n i c
presently
formula
coordinates
be mentioned
coordinates
made quantitively
techniques
the
of harmonic
It s h o u l d
of harmonic
is t h a t
any derivatives
discussion
[JK2].
and were
sions,
for z _~ ~ < 2~
for this phenomenon
properties [dTK]
o~< 1)
for ~ < ~ _~ 2~
O
A more
(0
O suf-
.
find a conformal
interior
(10.2.10)
choosing
to D
map k satis-
.
imply
=< c 6
10.2.8),
we
infer
by a r e s u l t
from
linear
elliptic
the-
ory
(10.2.17)
IgiJlcl, ~ < c 7
(In o r d e r that
to a p p l y
the C e - ~ o r m s
tor A on B ordinates i.e.
J(h)
of the
are b o u n d e d .
o on B ° g i v e n
a C I -bound
(g13),
this
and
on
result
from elliptic
coefficients This
by h
, for w h i c h
g:
= det(glj)
we have
the L a p l a c e
is no p r o b l e m ,
the c o e f f i c i e n t s
for w h i c h
of
theory,
we
since
already
of the
is c o n t r o l l e d
-Beltrami
we
by
opera-
can use the co-
proved
inverse
to e n s u r e
(10.2.16),
metric
tensor
the a s s u m p t i o n
~ 6 -I )
Since
gij
= < grad h i
, grad h j>
, (10.2.17)
implies
(10.2.1).
q.e.d.
10.3.
Bounds
Corollary , has
on
10.1:
Suppose
a boundary
ordinate
map
~B of
B,a bounded
class
lhlc2,~ < c 7
c 7 = c7(e,= ~
If
, then
n-Woo
-I x 6 u° (Z2~B)
Since
the
implies
PnOUnOFn
are
sufficiently
Since sets
on of
equal
to
~
on
BD
, Po(X)
~ > 0
large
other
x 6 ~I ~ ~
also
n
.
hand,
D , this would
sumption We
and
that
d ( F o l (x),SD)
for
equicontinuous
the
F c o n v e r g e u n i f o r m l y to F on c o m p a c t s u b n x £ F(D) = Q w h i c h c o n t r a d i c t s the as-
imply
" This
proves
(11.2.4)
have
-l(z2 ~Bo~ =uolc~B~ u uol(z2-~>
Uo and
since
joint, (SB)
the
we
sets
uol ( ~ B )
can assume
vanishes
for
cover
w.l.o.g,
one
a neighborhood
that
chosen
~
the
of x ° and
twodimensional
are
dis-I uO
measure
of
' there
exists
. If
6 uol ( E 2 ~ B ) ,
x
then
lim Pn(X)
= Po(X)
>
n-~oa
and
because
an o p e n n
. This
o Therefore follows
of
the equicontinuity
neighborhood
U of x such
of
the
that
functions
p n IU > ~
Pn
for sufficiently
large
implies
= lim n+oo
~
n
= lira u n+~
n
u° = Uo almost from
(11.2.4).
By
= u
o
on
U
everywhere the
choice
.
on of
Uo I(~2~B uo
), a n d
, we have
on
(11.2.3) the other
now hand
111
EE
(uo)
~ EZ
I Thus,
(~o) I
we
conclude
from
(11.2.2)
and
(11.2.3)
that
E 2 ( ~ o ) = E ~ ( u o) and
consequently
E D ( V O)
Since
v
energy on D
= ED(UoOF)
a n d u oF c o i n c i d e o
o
minimizing
. Therefore
phism,
the
chosen
point
11.3. With
Theorem
of Theorem
11.2: ~
a n d 5.1)
f r o m the
that v
also u
o
finishes
the p r o o f
exists
can a l s o
~ and
class
a harmonic
to
among
all d i f f e o m o r p h i s m s
satisfies
and
of
of
u oF c o i n c i d e o diffeomor-
an a r b i t r a r i l y
of Theorem
11.1.
8.1 improve
its i m a g e
topic
o
uniqueness
is a h a r m o n i c
is a n e i g h b o r h o o d
of Lipschitz
of ~ onto
are o f L i p s c h i t z there
conclude
Thm.
8.1
L e t ~ c ZI b e a t w o d i m e n s i o n a l
consisting
homeomor~hism
Then
4.1
, which
we
, we
consequently
. This
same method,
boundar~
~(~)
in ~
x ° 6 ZI
~D
(Thms.
u oF a n d o
latter
Extension the
maps
on
curves,
~(~),
and suppose
and convex with diffeomorphism
u = ~ on
~
domain with
nonempty
a n d l e t 4: ~ ÷ Z 2 b e that
respect
to ~ ( ~ ) .
u: ~ ÷ ~(~)
. Moreover,
a
the c u r v e s
which
u is o f
homotopic
to ~ a n d a s s u m i n g
[JS].
case
is h o m o -
least energy
the s a m e b o u n -
dary values. This
result
ture was
Proof:
rise
strictly In t h i s
that
Remarks
respect
an o b v i o u s
involving
in
first
the p r o o f
arguments
As s h o w n
from
The
image
curva-
~
Thm.
~
are of
class
and ~(~)
C 2+~
and that
and that ~(~)
is
to ~(~).
proceeds change 7.2.
an d ~ ( ~ ) between
along of
The
the
the
lines
o f the p r o o f
replacement
general
case now
argument follows
of Theo-
at boundaby approxi-
as in 8.2.
about
7.4,
of non - positive
[SY1].
convex with case,
ry p o i n t s
11.4.
in
to a d i f f e o m o r p h i s m
11.1 w i t h
mation
taken
We assume
gives
rem
is
solved
the
situation
in higher
in h i g h e r
dimensions
one
dimensions
cannot expect
an a n a l o g u e
of
112
Thm.
11.1 or even of Cor.
the image was gative
flat,
curvature
still
On the o t h e r hand,
11.1
however,
remains
carrying
structure,
it was p o s s i b l e
cf.
a complex
[JY].
to be rather answer only
K~hler m a n i f o l d s s uita b l e m e t r i c
12.1.
Holomorphicity
In
and thus
ne-
i.e.
Riemannian
its
Riemannian
that the
(unique)
a diffeomorphism,
in these papers
the q u e s t i o n
seem
of the e x i s t e n c e
s e t t i n g has a s a t i s f a c t o r y
1 , since o n e - d i m e n s i o n a l orientable
conformal
compact
surfaces
with a
structure.
maps b e t w e e n
of certain h a r m o n i c
and E 2 are
Eells
surfaces
maps
closed orientable
of a surface
12.1:
and W o o d o b t a i n e d
Suppose
and an a n a l y t i c
surfaces,
Z , and d(~)
proof
X(Z)
of
denotes
is the degree
determined
of a
Theorem
topological
12.2:
surfaces,
to me-
relative
to the complex
struc-
. an a n a l y t i c a l
r e s u l t of H. K n e s e r
and furthermore
respect
,
and W o o d to give
Suppose
result:
If
or a n t i h o l o m o r p h i c
by y and g
12.1 e n a b l e d Eells
following
' resp.
IX(X2) I > 0
then h is h o l o m o r p h i c
the f o l l o w i n g
h: gl ÷ Z2 is h a r m o n i c w i t h
7 and g on ZI and Z2
x(x 1) + Id(h) l
Thm.
7.4,
.
Theorem
tures
of
strictly
Theorem
that ZI
[EWI],
trics
considered
but c o m p a c t
of h a r m o n i c
the E u l e r c h a r a c t e r i s t i c map ~
cases,
with
is n e c e s s a r i l y
in the K ~ h l e r
dimension
are n o t h i n g
Applications
Kneser's
class
and c o r r e s p o n d i n g
12.
Suppose
however,
diffeomorphism
in c o m p l e x
compatible
to show in some
The image m a n i f o l d s
special,
of a h a r m o n i c
in the example
the image has
of K ~ h l e r manifolds,
structure
map in a given h o m o t o p y
[Si] and
Since
open.
in the c o n t e x t
manifolds,
harmonic
to hold.
the case w h e r e
again
[Kn2]
that E I and Z2 are
X(Z 2) < O
. Then
p r o o f of the
closed o r i e n t a b l e
for any continuous
map
~: ~I ÷ ~2 (12.1.1) Proof
Id(~)IX(~ 2) ~ X(ZI)
of T h e o r e m
12.2:
We i n t r o d u c e
some metrics
y and g on ZI and
113
Z2
' resp.,
Thm.
12.1,
and
find a harmonic
h is
(anti)
This,
however,
which
says
map h
. Therefore,
map h homotopic
holomorphic
in case
is in c o n t r a d i c t i o n
[d(h)]x(Z 2) = X ( ~ I ) + (12.1.1)
to the
to ~ by Thm.
By
]d(~)IX(Z 2) < X(EI). Riemann - Hmrwitz
r , r ~ O for an
must
4.2.
(anti)
formula,
holomorphic
hold.
q.e.d.
Before
proving
Corollary m a p h:
Thm.
12.1:
ZI ÷ E2 is
22.1,
we n o t e
two o t h e r
__If ZI
is d i f f e o m o r p h i c
(anti)holomorphic
(and
interesting
consequences
to S 2 , then
any h a r m o n i c
therefore
constant,
if X(Z 2)
0). This
is
due
Corollar~z then
Cor.
12.2,
12.2.
and L e m a i r e
is no h a r m o n i c
due
map
Proof
In this
[WI]
map
h:
to E e l l s - W o o d , I is
of Theorem
section,
Z I + Z 2 with
torus,
d(h)
= ±I
and Z2 to S 2 , , for any m e -
follows
a covering
f r o m Thm.
12.1,
since
any h o l o -
map.
12.1
we want
use
to the
.
of d e g r e e
shall make
[LI].
I_~f E I is d i f f e o m o r p h i c
on Z I a n d Z 2
morphic
We
12.2:
there
trics
to W o o d
to p r o v e
of some
Thm.
computations
12.1. of S c h o e n
and Yau
[SYI]
in the
sequel. It is c o n v e n i e n t If p 2 ( z ) d z d ~ nate
charts
to use
map
h -- + 2 ~ h ZZ
Lemma
complex are
12. 1:
O
cf.
Thm.
At points,
, cf.
conformal
where
~h o_~r ~h, r e s p . ,
(12.2.3)
Alog
the G a u s s
curvature
is n o n
_ f -hi 2)
l[hr2 = K I + ~2(l~hl 2 of Z, l
coordi-
on E I and E 2 , resp.,
(1.3.4)
AJ_og [~h[ 2 = ~ , using
(12.2.6)
P I
3z
= ±2
~(", (~)
P
z
, h, (~)
,
hz
P
p where
R denotes
the c u r v a t u r e
3%-
t e n s o r of E
=-K213hl2(j(h))
~p < D ~ h z
2
+
D~_ ~ ~
>
, %)
3-'~
+
115
where
J(h)
=
I~hl 2 -
l~-hl2
is the J a c o b i a n
of h
.
Mo re ove r,
(12.2.8)
I _~_ ~z
O}
is a c o m p a c t
( 7. ) = M( Z )/D( 7. ) . S u p p o s e
distinct
(7)
follows.
exists
. By Lemma
bounded
ui(~i,g)
y in R
of Z
curvature
to a s s u m e
, (12.4.2)
, there
a region
{p £ T(Z):
moduli
y on ~
identity
constant us
dr . 2 r sln
drd~2 ~ 2E(u) r sin ~0
0 ÷ 0 as 1 + 0
B y Thm.
By
contains
< e I , 0 < arg
I/2 )
arclength
12 (~ - 20 ) ~-8 o 1 ~ S 8
X
(Z,y)
-
(12.4.3)
and
(7.,X)
, we have
(12.4.3) where
and
to the i d e n t i t y .
[Hp],
the P o i n c a r ~ u p p e r h a l f p l a n e , i d e n t i f y i n g 1 z + e z , a n d e + 0 as 1 + 0 . Since
-I
,
is a h a r m o n i c
of Keen
curvature
Y i is a s e q u e n c e
all correspond
projection
T
subset
on the other hand
to t h e
( 7.)+ R ( Z )
o f the that
of mutually same point = T(Z
)/
123
(D( Z ) / using
Do(Z)).
By e q u i c o n t i n u i t y
the fact that all 7i h a v e
constant
curvature
seque n c e w h i c h homotopy
classes1~y
proper
function a compact
Theorem,
of Yi" These
curvature
metric),
+ 7, ui(Yi,g) ÷ u ( 7 , g )
radius
we can choose
• u is h a r m o n i c w i t h
by c o n t i n u i t y of E as a function of y . I) w h e n c o m p o s e d w i t h the c o n f o r m a l and h e n c e Theorem
12.6 :
Proof: We
T (Z) is t o p o l o g i c a l l y
(A. T r o m b a
fix any c o n f o r m a l
g with nonpositive m o n i c map u(y,g) E(u(y,g)) Ever y
curvature.
homotopic
point
by Thm.
12.4,
fore of
d e g r e e one,
Therefore,
convergent
sub-
that E is a ui(Yi,g)
again
stays
(by M u m f o r d ' s
bounded below
a convergent respect
for
subsequence
to y an m i n i m i z i n g
ener~,y p r e s e r v i n g
map y1 + Y i
a cell.
[Tr]): structure
as a f u n c t i o n
critical
sequence
uniformly
11.2),
for their
are in d i f f e r e n t
imply
( Z ). By e q u i c o n t i n u i t y
injectivity
3.3 or
radius
(Yi,g)
assertions
on T( Z ) and that a m ~ n i m i z i n g region of T
(cf.
a uniformly
the fact that all u i
choice
all Yi now have
their c o n s t a n t 7i
metric) , the u i (yi, g) have
contradicts
within
of the ui(Yi,g)
the same i n j e c t i v i t y
p on Z and r e p r e s e n t For every m e t r i c
to the identity.
of the c o n f o r m a l
gives
and since
rise
u(y,g)
no b r a n c h
We now
structure
to a c o n f o r m a l is h o m o t o p i c
points
it by some m e t r i c
y , we can look
find a har-
at E =
represented
branched
by y .
cover u(y,g),
to the i d e n t i t y
and there-
occur.
the global m i n i m u m p of Thm.
12.5 has to be the only
criti-
cal p o i n t of E on T(Z). It is s t r a i g h t f o r w a r d p is p o s i t i v e and the result
12.5.
The
Still
another
outlined
tional
definite follows
approach
(cf.
extremal Given
a n d Rauch
two
metric
the
harmonic
finally
zing map on the m e t r i c tional p r o c e d u r e
complex
to T e i c h m ~ l l e r 1954.
structures
class
They
maps by s o l v i n g
minimum,
tried
these
conformal
dependance
to obtain
integral
and then m a x i m i z e
a continuous
theory w a s
on a surface,
of maps b e t w e e n
map o v e r all such
varia-
they m i n i structures
the e n e r g y metrics
of
with
of the m i n i m i -
to show that a r e s u l t of such a t w o f o l d v a r i a -
is an e x t r e m a l
of a s o l u t i o n
maps
[GR] in
on the image
assume
of the first v a r i a t i o n a l existence
of E at
Hence p is a n o n d e g e n e r a t e
quasiconformal
for any c o n f o r m a l corresponding
variation
theory.
of h a r m o n i c
in a given h o m o t o p y
fixed area and
that the s e c o n d
of G e r s t e n h a b e r - R a u c h
application
problems.
energy
[Tr]).
from Morse
by G e r s t e n h a b e r
TeichmHller's
mize
to c a l c u l a t e
problem
quasiconformal is p r o v i d e d
to the s e c o n d
map.
While
by L e m a i r e ' s
one is not clear,
the s o l u t i o n Theorem,
the
and the conti-
124
nuous
dependance
is n o t k n o w n fore,
this
is a l s o n o t k n o w n
for
is
an a r b i t r a r i l y
an i n t e r e s t i n g ,
because
curved
but
uniqueness
of harmonic
image metric,
unfortunately
cf.
still
5.5.
maps
There-
incomplete
pro-
gram.
12.6.
Harmonic
L e t F:
X ÷ ~n+p
into ~n+p G(p,n)
denote
. The
Gauss
of p - p l a n e s
plane Ruh
Gauss maps
and Vilms
mersed
[RV]
12.7:
an i m m e r s i o n map
theorems
o f the n - d i m e n s i o n a l
G: X ÷ G(p,n)
in ~ n + p
assigns
into
to e a c h
manifold
the G r a s s m a n n i a n point
X
manifold
x 6 X the normal
G: X ÷ G(p,n)
12t4:
H i_n_n~ 3 . T h e n
is
type
rallel
the G a u s s
under
12.4,
theorems
certain
is s h o w n
we
do n o t w a n t
to
[HJW]
12.7.
Surfaces
results
on
arises
from
surface, can b e
mean
curvature
into
image.
in t u r n
in
Gauss map Bern-
with
approach
This means
pa-
proves
a
manifolds
that the Gauss
that the correspon-
of ~n+p but
in this
curvature
(immersed)
This
implies
this point,
the
to p r o v e
Grassmannian
subspace
results
then
used
or submanifolds
space.
maps the
which
Gauss
field.
of constant
submanifolds
to e l a b o r a t e
refer
the
reader
in-
direction.
3- space
surfaces
of constant
Gauss
cur-
3 - space.
F is a s u r f a c e
= h11du2
is p o s i t i v e parameters positive
a minimal
These
[HOS] f o r s o m e
with
K o f F is p o s i t i v e ,
II:
Z is
is a l i n e a r
of constant
example
Suppose vature
and
i f X is i m -
X ÷ S 2 is h a r m o n i c .
for h a r m o n i c
submanifold
stead
in
G:
restrictions
Here,
vature
i.e.
to be c o n s t a n t
immersed
Another
map
if and only
curvature
field of Euclidean
theorem size
mean
for m i n i m a l
curvature
type
is h a r m o n i c ,
E is a s u r f a c e
antiholomorphic.
mean
Liouville
parallel
Suppose
If H = O in Cor. actually stein
proved
into ~n+p with
Corollary
ding
Bernstein
at F(x).
Theorem
map
and
+ 2h12dudv
definite x,y
([Db],
the s e c o n d
of
u,v
x,y are §725)
u,v
. If the G a u s s
fundamental
cur-
form
+ h22dv2
and can hence
instead
function),
and Darboux
local coordinates then
be
diagonalized
by
(i.e. II = l ( d x 2 + dy2),
called
discovered
isothermal that
introducing where
conjugated
u a n d v as
new
I is a
parameters,
functions
of x and
125
y satisfy
a s y s t e m of e l l i p t i c
fundamental not
form of F , i.e.
depending
extensively embedding
equations
on the i m m e r s i o n
by H. Lewy
depending
only on i n t r i n s i c of F into
only on the
geometric
3 - space.
[Lw] a n d E. Heinz
([Hz2])
This
first
quantities, fact was
for s o l v i n g
used
the Weyl
problem.
The s y s t e m
discovered
(12.7.1)
An +
by D a r b o u x
is the
(FII + I/2 ~
(log K ) ) ( u ~
+ (2r12 + i/2
~
I + F22
(v
+ u~) +
clog K))(ux
v2 )
+
following
Y
÷ UyVy
÷
= O
u 2 + u2 ) + 2 Av + F11 ( x y
+ (2r~2 ÷ I/2 ~
Clog K))(UxV x + UyVy) +
+ (r22 + I/2 ~ We see in p a r t i c u l a r constant,
that,
Clog~))(v~+
if the Gauss
then the t r a n s f o r m a t i o n
v ) = 0
curvature
(x,y) ÷
(u,v)
K of F is a p o s i t i v e
is harmonic.
(This
fact was p o i n t e d out to me by S. Hildebrandt). As an application, orem of L i e b m a n n
Theorem
12.8:
positive radius
a proof
[BI], p.
K into
3-space
By the Gauss - B o n n e t fundamental
x,y on the sphere
h:
By Cor.
S 2 which
a harmonic
S2 ÷ F
the-
F of constant sphere of
Theorem,
F is t o p o l o g i c a l l y
form of a given
diagonalize
immersion
theorem this
a sphere.
of F is p o s i t i v e
to o b t a i n
parameters
form. Since K { const.,
we
map
.
12.1, h is
the first
a closed surface
is given by a s t a n d a r d
we can use the u n i f o r m i z a t i o n
thus obtain
f o l l o w i n g well - k n o w n
.
the s e c o n d
definite,
of the
195).
The only i m m e r s i o n o f
curvature
I//K
Proof: Since
we p r e s e n t (cf.
(anti)
fundamental
form are p r o p o r t i o n a l
conformal
form.
Hence
everywhere,
and therefore
the
also d i a g o n a l i z e s
first and the s e c o n d
which means
fundamental
that the given immer-
126
sion
is e v e r y w h e r e
umbilical
and therefore
a standard
sphere
(cf.
[BI],
p.97) .
If we note
that in the s p h e r e - c a s e ,
and the T h e o r e m of R i e m a n n - Roch, is m u c h
in the s p i r i t of H. Hopf's
sphere
into
3 - space w i t h
sphere
(cf.
[Ho]).
constant
Cor.
12.1
follows
from L e m m a
then we see that the p r e c e d i n g proof,
that every
mean c u r v a t u r e
immersion
is a s t a n d a r d
1.1 proof
of a
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[ES]
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