Publ. Ma . 45 (2001), 387–397
A CARTAN-TYPE RESULT FOR INVARIANT
DISTANCES AND ONE-DIMENSIONAL HOLOMORPHIC
RETRACTS
Colum Wa
Abs ac
We de i e condi ions unde which a holomo phic mapping o a
au Riemann su ace mus be an au omo phism. This is an ana-
logue in ol ing in a ian dis ances o a esul o H. Ca an. Using
simila me hods we p o e an exis ence esul o 1-dimensional
holomo phic e ac s in a au complex mani old.
1. In oduc ion
In wha ollows, Wdeno es a connec ed Riemann su ace, Hol(W, W )
deno es he se o holomo phic sel -maps o Wand Au (W) deno es he
se o biholomo phic maps o Won o i sel . A complex mani old M
is au i and only i o each complex mani old N, each sequence o
holomo phic mappings om N o Mcon ains a subsequence which ei he
con e ges uni o mly on compac subse s o No is uni o mly di e gen
o infini y (in he one poin compac ifica ion o M) on compac subse s
o N.
Defini ion 1.1. We call a dis ance unc ion don a Riemann su ace W
in a ian i
d( (w), (z)) ≤d(w, z)∀w, z ∈W, ∀ ∈Hol(W, W ).
A He mi ian me ic hon Wis called in a ian p o ided
h( ∗(u),
∗(u)) ≤h(u, u)∀u∈O
wW, ∀w∈W, ∀ ∈Hol(W, W ).
We say ha a dis ance unc ion don Wis Ck(k≥1) i dis he in eg a ed
dis ance unc ion associa ed o a Ck−1He mi ian me ic hon W.
2000 Ma hema ics Subjec Classifica ion. 32F45, 32Q40, 53C60.
Key wo ds. Tau complex mani old, in a ian dis ance, au omo phism, holomo phic
e ac .
388 C. Wa
Rema k 1.A s anda d example o an in a ian me ic is he squa e o
he Kobayashi me ic on a au Riemann su ace. Fo he uni disc in he
complex plane, his me ic is usually e e ed o as he Poinca ´e me ic.
In his pape [4], J. P. Vigu´e p o ed he ollowing esul .
Theo em. Le Xand Wbe connec ed, au complex mani olds wi h W
1-dimensional. Assume ha his an in a ian He mi ian me ic on W
and choose w∈W,x∈Xand ∈O
xX {0}. Then he e exis s a
holomo phic e ac ion ρ:X→Xsuch ha ρ(w)=x,ρ∗(OwW)=C
and ρ(X)is biholomo phic o Wi and only i E(x, )=F(x, )(whe e
E,F :OX→R+a e in a ian Finsle me ics which a e defined in
e ms o w,W,h,xand Xand which gene alise he usual Ca a h´eodo y
and Kobayashi me ics).
Key o his p oo o his is he ollowing esul o H. Ca an [1].
Ca an’s Theo em. Le Wbe a au Riemann su ace. I ∈Hol(W,W )
fixes some poin w∈Wand has unimodula de i a i e a w hen ∈
Au (W).
No e ha he hypo hesis o Ca an’s esul is equi alen o he equi e-
men ha fixes wand ha i s de i a i e a wis uni a y wi h espec
o e e y (in pa icula e e y in a ian ) He mi ian me ic on W.Ina
ema k in [4], J. P. Vigu´e defined he in a ian pseudodis ances cW,w
X,x
and kW,w
X,x in e ms o an in a ian dis ance on W(see Sec ion 3 be-
low). He would ha e liked o use hese o in es iga e he exis ence o
1-dimensional holomo phic e ac s h ough wo gi en poin s o X.To
do so, Vigu´e would ha e needed (bu did no possess) an analogue o
Ca an’s esul which uses an in a ian dis ance in place o an in a ian
He mi ian me ic. In Sec ion 2 we p o e such an analogue (Theo em 2.3)
o Ca an’s heo em unde he assump ion ha he in a ian dis ance
a ises om a con inuous He mi ian me ic on W. Then in he final
sec ion we use he in a ian pseudodis ances cW,w
X,x and kW,w
X,x (which gen-
e alise he usual Ca a h´eodo y dis ance and Kobayashi unc ion on a
complex mani old) and apply Theo em 2.3 o in es iga e he exis ence
o holomo phic e ac ions o a complex mani old on o a 1-dimensional
submani old h ough wo gi en poin s.
2. Au omo phisms o Riemann su aces
Fi s we ecall some s anda d no ions om diffe en ial geome y. Le
hbe a con inuous He mi ian me ic on a connec ed Riemann su ace W.
Thus hde e mines a sesquilinea , posi i e defini e inne p oduc hw
In a ian Dis ances and Holomo phic Re ac s 389
on each angen space OwWand hw a ies con inuously wi h w. The
associa ed no m on OwWis deno ed |·|
w( o simpli y no a ion, he
subsc ip wis o en omi ed). I ∈Hol(W, W ) we deno e i s de i a i e
a wby
∗w:OwW→O
(w)W.
The ope a o no m o ∗w(wi h espec o |·|
wand |·|
(w)) is deno ed
|| ∗w|| (o by || ∗|| when wis clea om he con ex ). As OwWis one-
dimensional i ollows ha
| ∗w(u)| (w)=|| ∗w||·|u|w∀u∈O
wW.
Con inui y o himplies ha he map w→ || ∗w|| is con inuous. A
piecewise C1pa h in Wis a mapping γ:[a, b]→W o which he e
exis s a fini e se o poin s a= 0<
1<···<
n=bsuch ha γ|[ i, i+1]
is C1and has nowhe e anishing angen o each i=0,... ,n−1. The
leng h o such a pa h γis defined by
l(γ)=b
a
h(γ( ),γ( ))1
2d =b
a
|γ( )|d .
Fo any wo poin s wand zin W he dis ance d(w, z) is defined by
d(w,z) = in {l(γ):γis a piecewise C1pa h om w o z}.
This unc ion is clea ly symme ic, posi i e and sa isfies he iangle
inequali y. I is a s anda d esul ha d(w,z)>0 when w=zand ha
dgene a es he gi en opology on W( o example, see [2]). The open
ball B(w, )⊂Wwi h cen e wand adius >0 is gi en by
B(w, )={z∈W:d(w, z)< }.
I dis he dis ance a ising om a he mi ian me ic h, i is easy o show
ha in a iance o himplies in a iance o d. In his fi s p oposi ion we
p o e a con e se esul .
P oposi ion 2.1. Le dbe he in eg a ed dis ance associa ed o a con-
inuous He mi ian me ic hon a Riemann su ace W.I dis in a ian
hen hmus also be in a ian .
P oo : Assume ha he e exis s ∈Hol(W, W ) such ha || ∗w|| >1 o
some w∈W. We will show ha dcanno be in a ian .
As || ∗|| =0a w, maps some neighbou hood Uo Wbiholomo -
phically on o an open neighbou hood o (w). Sh inking Ui necessa y
and using con inui y, we may assume ha || ∗|| ≥ 1+ on U o some
>0.
390 C. Wa
Choose δ>0 such ha B( (w),δ)⊂ (U). Le y∈B( (w),δ) and
le γbe a pa h om (w) oysuch ha
d( (w),y)≤l(γ)<δ.
As any pa h which lea es B( (w),δ) will ha e leng h a leas δ, he image
o γmus be con ained in B( (w),δ). Thus we may w i e γ= σ whe e
σ=( |U)−1γlies in Uand s a s a w. Deno e he endpoin ( |U)−1(y)
o σby z. Then
l(γ)=|γ( )|d
=|( σ)( )|d
=|| ∗σ( )||·|σ( )|d
≥(1 + )|σ( )|d
≥(1 + )d(w,z).
Taking he infimum o e pa hs joining (w) oy= (z), i ollows ha
d( (w), (z)) ≥(1 + )d(w, z).
Hence dcanno be in a ian .
P oposi ion 2.2. Le dbe an in a ian C1dis ance on a Riemann su -
ace W.Le ∈Hol(W, W )and assume ha he e exis s a sequence o
pa hs γn:[0,a
n]→Wwhich all s a a wand o which
lim
n→∞ l(γn) = lim
n→∞ l( γn)=a>0.
Then || ∗w|| =1.
P oo : In a iance o dimplies ha || ∗|| ≤ 1 e e ywhe e. Assume ha
|| ∗w|| = <1. As dis C1,w→ || ∗w|| is con inuous and hence he e
exis s >0 such ha || ∗|| <1+
2on B(w, ). Fo each ndefine n∈
(0,a
n]by
n=ani γn([0,a
n]) ⊂B(w, )
sup{ :γn([0, ]) ⊂B(w, )}o he wise.
In a ian Dis ances and Holomo phic Re ac s 391
Then
l γn|[0, n]= n
0
|( γn)( )|d
= n
0
|| ∗γn( )||·|γ
n( )|d
<1+
2 n
0
|γ
n( )|d
=1+
2lγn|[0, n]
and hence
l( γn)=l γn|[0, n]+l γn|[ n,an]
<1+
2lγn|[0, n]+lγn|[ n,an]
=l(γn)−1−
2lγn|[0, n]
≤l(γn)−1−
2min( ,l(γn)) ∀n.
Thus
lim
n→∞ l( γn)≤a−1−
2min( ,a)<a since a>0.
As his con adic s he hypo hesis ha l( γn) con e ges o a, ou as-
sump ion ha || ∗w|| <1 mus ha e been alse.
We combine hese wo p oposi ions o p o e he ollowing heo em.
Theo em 2.3. Le dbe a C1in a ian dis ance on a au Riemann su -
ace W. Assume ha ∈Hol(W, W )and ha he e a e dis inc poin s
wand zin Wsa is ying
(w)=wand d(w,z)=d(w, (z)).
Then ∈Au (W).
P oo : Le γnbe a sequence o pa hs om w o zwhose leng hs con e ge
o d(w, z). Taking he limi as n→∞in he inequali y
l(γn)≥l( γn)≥d( (w), (z)) = d(w, z)
we deduce ha
lim
n→∞ l(γn) = lim
n→∞ l( γn)=d(w,z)>0.
392 C. Wa
P oposi ion 2.2 now implies ha || ∗w|| = 1. Since wis fixed by
and Ow(W) is one dimensional, ∗wmus be gi en by mul iplica ion
by a unimodula complex numbe . Ca an’s heo em now implies ha
∈Au (W).
Co olla y 2.4. Le Wbe a au Riemann su ace and suppose ∈
Hol(W, W )fixes wo dis inc poin s o W. Then ∈Au (W).
P oo : The map p ese es he Kobayashi dis ance be ween he wo
fixed poin s. As he Kobayashi dis ance is C∞( o any au Riemann
su ace) he p eceding heo em implies ha ∈Au (W).
Co olla y 2.5. I ∈Hol(W, W )fixes wo dis inc poin s w,z ∈W
which can be joined by a unique pa h γ:[0,a]→Wsa is ying
d(w,z)=l(γ)and l(γ|[0, ])= ∀
hen is he iden i y map.
P oo : The pa h γ also joins w o z. Fo any ∈[0,a]weha e
d(w,z)=l(γ|[0, ])+l(γ|[ ,a])
≥l( γ|[0, ])+l( γ|[ ,a]) by in a iance
≥d( (w), (z))
=d(w,z).
I ollows ha we mus ha e l(γ|[0, ])=l( γ|[0, ]) o all . Ou unique-
ness hypo hesis o γnow implies ha (γ( )) = γ( ) o all ,so
fixes each poin on γ([0,a]). The iden i y heo em o analy ic unc ions
implies ha is he iden i y map.
Rema k 2.I he dis ance dis C4 hen each poin w∈Whas a neigh-
bou hood Usuch ha any wo poin s o Ucan be joined by a unique
pa h in Uwhich sa isfies he hypo hesis o γin he p eceding co olla y
(such pa hs a e usually called leng h minimising geodesics). Fo a au
Riemann su ace W, he Kobayashi dis ance is C∞and hence any holo-
mo phic map ∈Hol(W, W ) which fixes wo sufficien ly close poin s
mus be he iden i y mapping.
Rema k 3.The biholomo phism w→ 1
won he annulus A={w∈C:
1
2<|w|<2}fixes he wo poin s 1 and −1. Howe e he e is mo e
han one leng h minimising geodesic joining hese wo poin s in A(wi h
espec o he Kobayashi me ic).
In a ian Dis ances and Holomo phic Re ac s 393
3. One-dimensional holomo phic e ac s
In his sec ion, ollowing J. P. Vigu´e[4], we define analogues o he
Ca a h´eodo y pseudodis ance and he Kobayashi unc ion on a complex
mani old. We use hese o examine he exis ence o holomo phic e ac-
ions o a complex mani old on o a 1-dimensional complex submani old
h ough wo gi en poin s.
Le x1,x
2,... ,x
nand y1,y
2,... ,y
nbe o de ed sequences o poin s
in he complex mani olds Xand in Y espec i ely. Then
Hol(X, x1,... ,x
n,Y,y
1,... ,y
n)
= ∈Hol(X, Y ): (xi)=yi∀i=1,... ,n
.
Le Wbe a connec ed Riemann su ace and dan in a ian dis ance on
W. Fix a poin w∈Wand le Xbe a complex mani old wi h basepoin
x∈X. Then we define a Ca a h´eodo y ype unc ion on X×Xwi h
alues in [0,∞]by
cW,w
X,x (x1,x
2) = supd( (x1), (x2)) : ∈Hol(X, x, W, w)∀x1,x
2∈X.
The Kobayashi e sion kW,w
X,x (x1,x
2) is defined as ollows
(i) I Hol(W, w, w1,w
2,X,x,x
1,x
2)=∅ o all w1and w2in W hen
kW,w
X,x (x1,x
2)=∞.
(ii) O he wise kW,w
X,x (x1,x
2) is gi en by
in d(w1,w
2):w1,w
2∈W, ∈Hol(W, w, w1,w
2,X,x,x
1,x
2).
I ollows om he in a iance o d ha
cW,w
X,x (x1,x
2)≤kW,w
X,x (x1,x
2)∀x1,x
2.
Fo he special case (X, x)=(W, w), he in a iance o dalso implies
cW,w
W,w(w1,w
2)=kW,w
W,w (w1,w
2)=d(w1,w
2)∀w1,w
2∈W.
As in he cases o he usual Ca a h´eodo y and Kobayashi unc ions i is
s aigh o wa d o show ha o all ∈Hol(X, x, Y, y) and x1,x
2∈X
cW,w
X,x (x1,x
2)≥cW,w
Y,y ( (x1), (x2))
and
kW,w
X,x (x1,x
2)≥kW,w
Y,y ( (x1), (x2)).
I we ake he usual Kobayashi dis ance on Was ou in a ian dis ance d,
hen i is easy o see ha he esul ing unc ion kW,w
X,x sa isfies
kW,w
X,x ≤k
394 C. Wa
whe e kdeno es he usual Kobayashi unc ion gi en by
k(x, x1) = in anh−1
z−w
1−wz
:∃w,z ∈D
wi h Hol(D,w,z,X,x,x
1)=∅
whe e Ddeno es he uni disc in he complex plane.
We now use he unc ions cW,w
X,x and kW,w
X,x o gi e a c i e ion o decid-
ing when he e exis s a holomo phic e ac ion o a complex mani old X
on o a submani old biholomo phic o Wwhich passes h ough wo gi en
poin s o X. Fi s we ecall he defini ion o a holomo phic e ac .
Defini ion 3.1. Aholomo phic e ac ion o Xis a holomo phic map-
ping ρ:X→Xsuch ha
ρ|ρ(X)is he iden i y map on ρ(X).
The se ρ(X) is called a holomo phic e ac o X. I is closed and
analy ic.
P oposi ion 3.2. Le xand x1be dis inc poin s in a complex man-
i old Xand le dbe an in a ian dis ance on a Riemann su ace W.
Assume ha he e exis s a holomo phic e ac ion ρ:X→Xsuch ha
x, x1∈ρ(X)and ρ(X)is biholomo phic o W. Then he e exis s some
poin w∈W o which
0<c
W,w
X,x (x, x1)=kW,w
X,x (x, x1)<∞.
P oo : Le i:W→Xbe a biholomo phism o Won o ρ(X). Pu w=
i−1(x) and w1=i−1(x1). The h ee inequali ies
(i) d(w,w1)≥kW,w
X,x (i(w),i(w1)) = kW,w
X,x (x, x1)
(ii) cW,w
X,x (x, x1)≥d(i−1ρ(x),i
−1ρ(x1)) = d(w,w1)
(iii) cW,w
X,x (x, x1)≤kW,w
X,x (x, x1)
combine o gi e
0<d(w, w1)≤cW,w
X,x (x, x1)≤kW,w
X,x (x, x1)≤d(w,w1)<∞
since w=w1. The esul ollows.
By s eng hening ou hypo heses, we can p o e he ollowing con e se
o his p oposi ion.
In a ian Dis ances and Holomo phic Re ac s 395
Theo em 3.3. Le xand x1be dis inc poin s in a connec ed, au com-
plex mani old X.Le dbe a C1in a ian dis ance on a au Riemann
su ace W. I he e is some poin w∈W o which
(a) cW,w
X,x (x, x1)=kW,w
X,x (x, x1)<∞and
(b) he open ball B(w, )has compac closu e in W(whe e =kW,w
X,x (x,x1)),
hen he e exis s a holomo phic e ac ion ρ:X→Xsuch ha x, x1∈
ρ(X)and ρ(X)is biholomo phic o W.
P oo : Assume ha he e is a poin w∈Wwhich sa isfies he hypo he-
ses (a) and (b). By au ness o Xwe can find maps , 1,
2,... in
Hol(W, w, X, x) and a sequence o poin s zn∈Wsuch ha
(i) n→ uni o mly on compac se s,
(ii) n(zn)=x1 o each n,
(iii) lim
n→∞ d(w, zn)=kW,w
X,x (x, x1).
As B(w, ) is compac and Wis locally compac , he e exis s >0 such
ha B(w, + ) is compac . By (iii), he e exis s Nsuch ha zn∈
B(w, + ) o all n≥N. Compac ness o B(w, + ) implies ha zn
has a con e gen subsequence. Passing o his subsequence i necessa y,
we may assume ha zncon e ges o z(say). Since dis con inuous, we
ob ain
d(w, z) = lim
n→∞ d(w,zn)=kW,w
X,x (x, x1).(1)
As he se {z,z1,z
2,...}is compac , condi ions (i) and (ii) imply ha
(z) = lim
n→∞ n(zn)=x1.
No e ha zand wa e dis inc . O he wise we would would ha e x=
(w)= (z)=x1which con adic s he hypo hesis ha xand x1a e
dis inc .
Nex we use he au ness o W o cons uc a sequence gn∈
Hol(X, x, W, w) which con e ges uni o mly on compac se s ( o gsay)
such ha
lim
n→∞ d(gn(x),g
n(x1)) = cW,w
X,x (x, x1).
Le w1=g(x1). As (gn(x),g
n(x1)) con e ges o (g(x),g(x1))=(w, w1)
and dis con inuous, we ob ain
d(w,w1)=cW,w
X,x (x, x1).(2)
