This article was downloaded by: [Moskow State Univ Bibliote]On: 01 November 2013, At: 20:12Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK

Complex Variables, Theory and Application: AnInternational JournalPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/gcov19

Some classical theorems and families of normalmaps in several complex variablesJames E. Joseph a & Myung H. Kwack aa Department of Mathematics , Howard University , Washington, DC, 20059, USAPublished online: 14 Jan 2008.

To cite this article: James E. Joseph & Myung H. Kwack (1996) Some classical theorems and families of normal maps inseveral complex variables , Complex Variables, Theory and Application: An International Journal, 29:4, 343-362

To link to this article: http://dx.doi.org/10.1080/17476939608814902

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purposeof the Content. Any opinions and views expressed in this publication are the opinions and views of theauthors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should notbe relied upon and should be independently verified with primary sources of information. Taylor and Francisshall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, andother liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relationto or arising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Complex Variables, 19%, Vol. 29, pp. 343-362 @ 1996 OPA (Overseas Publishers Association) Reprints available directly from the publisher Amsterdam B.V. Published in The Netherlands Photocopying permitted by license only under license by Gordon and Breach

Science Publishers SA Printed in Malaysia

Some Classical Theorems and Families of Normal Maps in Several Complex Variables*

JAMES E. JOSEPH and MYUNG H. KWACK Department of Mathematics, Howard University, Washington, DC 20059, USA

Let H(X,Y)(C(X,Y)) represent the family of holomorphic (continuous) maps from the complex (topo- logical) space X to the complex (topological) space Y, and let Yt = Y u {m} be the Alexandroff one-point compactification of Y. We say that F c H(X,Y) is uniformly normal if {f o cp : f E F, cp E H(M,X)} is a relatively compact subset of C(M,Yt) (with the compact-open topology) for each com- plex manifold M. We show that normal maps as defined by authors in various settings are, as singleton sets, uniformly normal families, and prove theorems for uniformly normal families. These theorems in- clude (1) answers to questions posed by Hahn on higher dimensional formulations of classical theorems of Lehto and Virtanen and the Five-Point Criterion of Lappan, and (2) strong forms of Schottky's Theo- rem proved for and shown to characterize such families of functions. This last result is applied to present a higher dimensional version of a classical lemma of Bohr.

AMS No. 32A10, 32C10, 32H20, 32A17, 54C20, 54C35, 54D35, 54C05 Communicated: R. F? Gilbert (Received March, 1995)

INTRODUCTION

In 1957 Lehto and Virtanen defined a function f meromorphic on the complex unit disk D = {z E C : lzl < 1) to be normal if {f o cp : (o E A(D)) is normal in the sense of Montel, where A(D) is the group of conformal automorphisms of D. Although the study of such functions was initiated by Noshiro nearly two decades earlier, Lehto and Virtanen are credited with having propelled this study to the forefront of modern mathematical research. Indeed, from 1957 to the present the subject of normal maps has been studied continuously and intensively, resulting in an extensive development in the single complex variable context and in generalizations to several complex variables settings (see [24] and list of references in [I-3, 5-7, 9-12, 15, 201).

Adopting terminology of Hayman (see Remark 2.16) we say that a family F of holomorphic maps from a complex space X to a complex space Y is uniformly nor- mal if for each complex manifold M , the collection {f o cp : f E F, cp E H(M,X) ) is relatively compact in C(M, Y +) where H(X, Y) represents the space of holomorphic maps from the complex space X to the complex space Y, Yf = Y U {ca} represents the Alexandroff one-point compactification of the topological space Y, Y+ = Y if Y is compact, and C(X,Y) represents the space of continuous maps from the topolog- ical space X to the topological space Y. The topology used on all function spaces is the compact-open topology (see [I71 for topological notions and particulars).

Preliminaries in Section 1 include the equivalence in ( lo) which is used frequently in subsequent sections.

- - -

*In memory of Professor Elbert E Cox

343

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

20:

12 0

1 N

ovem

ber

2013

344 J. E. JOSEPH AM) M. H. KWACK

( lo) If X,Y are complex spaces then F c H(X,Y) is uniformly normal iff F o H(D, X ) is uniformly normal.

We study uniformly normal families on hyperbolic manifolds in Section 2. We show that normal maps as defined by authors in various settings are, as singleton sets, uniformly normal families and that, in these settings, the maps in uniformly normal families are normal. We give an example of a family of functions which is not uniformly normal in the framework of the original Noshiro-Lehto-Virtanen def- inition of normal maps, even though each of its members is a normal function. In this section we provide significant generalizations of theorems of Brody, Lohwater and Pornmerenke, Lehto and Virtanen, Hahn, and Zaidenberg. An analogue of a theorem of Aladro and Krantz is presented for uniformly normal families. We in- troduce the notion of Brody limit and sequence and establish that the following are equivalent if M is a hyperbolic manifold, Y is a complex space and F c H(M,Y).

(2") F is uniformly normal. (3") F o H(D, M ) is an evenly continuous subset of H(D, Y). (4") There is a length function E on Y such that ldf l E < 1 for each f E F. (5") There is a length function E on Y such that E(hn(0),dhn(O)) -t 0 for each

Brody sequence {h,) for F. (6") There is a length function E on Y such that E(hn(0),dhn(O)) -t 0 for each

Brody sequence {h,) for F with a Brody limit.

In Section 3, a number of results in Section 2 for uniformly normal families on hyperbolic manifolds are shown to carry over to uniformly normal families on ar- bitrary complex spaces; for example, (2"), (3"), (5"), and (6") are equivalent when M is a complex space. A Zaidenberg generalization (to complex manifolds) of a single variable version of Schottky's Theorem due to Hayman is extended to uni- formly normal families of functions on arbitrary complex spaces and provided with a simpler proof, leading to (7"), a several variables form of a classical lemma of Bohr.

(7") Let X be a complex space, let B c X , f E H ( X , C ) satisfy (1) B is bounded with respect to the pseudo-metric kx, (2) supxEB If (x)I > 1 and (3) 0 E f (B). Then there is an r > 0 independent of f such that

Lappan's five-point criterion for characterizing normal maps from D to the Riemann Sphere pl(C) is generalized to uniformly normal families from complex spaces into the complex projective space Pn(C).

(8") Let X be a complex space, let o be a collection of at least 2n + 3 hyper- planes in general position in Pn (C) and let A = U, T. Then F c H (X, Pn (C)) is uniformly normal iff the following two conditions hold for F; (1) sup{ldf oP(O)l : P E H(D,X), ~ ( 0 ) E u ~ f -'(A)) < m, and (2) Each degenerate Brody limit for F is constant.

In particular, if X is a hyperbolic manifold we have

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

20:

12 0

1 N

ovem

ber

2013

NORMAL MAPS 345

(9") Let X be a hyperbolic manifold, let a be a collection of at least 2n + 3 hyperplanes in general position in Pn(C) and let A = U, r. Then F c H(X,Pn(C)) is uniformly normal iff the following two conditions hold; (1) sup{Idf (PI : P E U, f - 1 ( ~ ) 1 < 00, and (2) Each degenerate Brody limit for F is constant.

Finally, a theorem of Cima and Krantz on the boundary behavior of normal maps from a bounded domain in Cn into the Riemann Sphere is generalized to uniformly normal families into arbitrary complex spaces. Although it is not pointed out in each case, many of the findings in this paper are new for normal maps.

1. PRELIMINARIES

Let M be a hyperbolic manifold. Let KM represent the infinitesimal form of the Kobayashi distance kM on M, that is

KM(p,v) = infir > 0 : p(0) = p, dp(0,re) = v for some p E H(D,M))

where p E M, v E Tp(M), the tangent space of M at p, dp is the tangent map for p and e is the unit vector 1 at 0 E D (see [21], pp. 88-94). If Y is a complex space and E is a length function on Y we denote by dE the distance function generated on Y by E [21]. The norm Idf I E of the tangent map for f E H(M, Y) with respect to E is defined by

(We will use simply ldf 1 and (df (p)( when no confusion may arise). If F c C(X,Y) we say that F is evenly continuow from p E X to q E Y if for

each U open in Y about q, there exist V, W open in X,Y about p,q respectively such that

{ ~ E F : f ( p ) ~ W ) c { f ~ F : f ( V ) c U ) .

If F is evenly continuous from each p E X to each q E Y we say that F is evenly continuous (from X to Y) [17]. We will have the occasion to rely on the following topological version of the Ascoli-Arzelh Theorem. It is readily derived from a - Kelley-Morse theorem (Theorem 7.21 in [17]) since F(x) = F(x) for each x E X and, under the hypothesis, F is evenly continuous iff F is evenly continuous (F(x) = {f (x) : f E F) , 2 = topological closure of A).

PROPOSITION 1.1 Let X be a regular local& compact space and let Y be a regular space. Then F c C(X,Y) is relatively compact in C(X,Y) ifS

(a) F is evenly continuous, and (b) F(x) is relative& compact in Y for each x E X.

If F c C(Y,Z) and G c C(X,Y) we write F o G for {f og : g E G, f E F).

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

20:

12 0

1 N

ovem

ber

2013

346 J. E. JOSEPH AND M. H. KWACK

DEFINITION 1.2 Let X and Y be complex spaces. A family F C H ( X , Y ) is uni- form& normal i f F o H ( M , X ) is relatively compact in C ( M , Y + ) for each complex manifold M . We say that f is a normal map if { f ) is uniformly normal.

It is clear from Definition 1.2 that each member of a uniformly normal family is a normal map in that setting. Example 1.3 exhibits that a family of normal functions might fail to be uniformly normal.

EXAMPLE 1.3 Define F c H(D, pl(C)) by F = { f, : n = 1,2,. . .) where f,(z) = l / n ( n z + 1). Then (f,(z)( 2 l / n ( n + 1) on D and f, is a normal map in the framework of Lehtc-Virtanen by the Picard Theorem. Define c p , E A(D) by p,(z) = (n3z + ( 1 - n2))/((1 - n2)z + n3). Then f, o cp,(O) ft 0 while f, o cp,(n-l - n-3) -t 0. Hence F is not uniformly normal.

The proofs of Propositions 1.4 and 1.5 are omitted since they are evident from Definition 1.2.

PROPOSITION 1.4 If M is a complex manifold and Y is a complex space and F C H ( M , Y ) is uniformly normal then F is relatively compact in C ( M , Y ).

PROPOSITION 1.5 If X and Y are complex spaces the following statements are equiv- alent for F c H ( X , Y ) :

1. F is uniformly normal. 2. If Z is a complex space and G c H ( Z , X ) then F o G is uniformly normal. 3. I f Z is a complex subspace of X then the family of restrictions of the elements of

F to Z is uniformly normal.

In Proposition 1.6 we establish that the requirement that all complex manifolds M satisfy the condition in Definition 1.2 may be replaced by a condition on D.

PROPOSITION 1.6 I f X, Y are complex spaces the following are equivalent for F C H ( X , Y ) :

1. F is uniformly normal. 2. F o H(D, X ) is uniformly normal. 3. F o H(D, X ) is relatively compact in C(D, Y + ). 4. The closure of F in H ( X , Y ) is uniformly normal.

Proof (1) + (2) + (3) and (4) + (1); from Propositions 1.5 and 1.4. (3) + (1). If (1) does not hold we may assume M = { p E Cm : JJpIJ < 1) and,

by Proposition 1.1, that F o M ( M , X ) is not evenly continuous from 0 E M to q E Y f . There are sequences {p,) in M - {0} , {f,) in F, (9,) in H ( M , X ) such that I l P, I I -t 0, fn 0 cpn(0) -t q and fn 0 cp,(pn) ft q. Define A, E H(D, X ) by &(z) = ~ ~ ( z p , / l J p , J I ) ; f, 0 L ( O ) -t q while fn 0 Xn([[pn[[) f t q. From Proposition 1.1, F o H ( D , X ) is not relatively compact in C ( D , Y f ) , so (3) does not hold and we complete the proof of (3) + (1).

(1) + (4). We show that, for any complex manifold M ,

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

20:

12 0

1 N

ovem

ber

2013

1 NORMAL MAPS 347

Let g E F n H ( X , Y ) and cp E H ( M , X ) . There is a sequence { fn) in F such that fn + g. it follows that fn o cp -t g o cp.

1 From Proposition 1.6 it is not difficult to see that some important classes of com- I plex spaces are defined by uniformly normal families. A complex subspace X of a

1 complex space Y is hyperbolically imbedded in Y if for p, q E X and p # q there are

, open sets V, W in Y about p, q respectively such that k x ( V fl X , W n X ) > 0, where I

i k x is Kobayashi's pseudo-distance on X [19].

i I EXAMPLE 1.7 Kiernan [18] has shown that a relatively compact complex subspace of

a complex space Y is hyperbolical& imbedded in Y i f f H ( D , X ) is relatively compact I in H(D,Y), i.e. iff H ( D , X ) is a uniform& normal subset of H(D,Y).

I EWLE 1.8 Royden [26] showed that a complex manifold M is hyperbolic iff H(D, M ) is evenly continuous and Abate [I] showed that M is hyperbolic iff H(D, M ) is relative& compact in C(D, M +). Hence H(D,M) is a uniformly normal family iff M is hyperbolic.

EXAMPLE 1.9 It is shown in [16] that a complex subspace X of a complex space Y 1 is hyperbolical& imbedded in Y i f f H(D, X ) is relatively compact in C(D, Y ' ) , i.e. i f f

1 H(D, X ) is a unifomly normal subset of H(D, Y ) . I

In Proposition 1.10 we show that families of distance decreasing maps between metric spaces are relatively compact as maps into the one-point compactifications of the ranges.

I PROPOSITION 1.10 Let ( Y , a ) be a locally compact metric space. Let X be a topolog- i ical space and let p be a pseudometric on X which is continuous on X x X . If each ( f E F c C ( X , Y ) is distance decreasing with respect to p and u (i.e. if a ( f ( x ) , f ( y ) ) 5 I

p(x,y) for all f E F , x,y E X ) then F is relatively compact in C ( X , Y + ).

Proof We show that F is evenly continuous from X to Y i . If not, there exist p E X , 9,s E Y f and nets {p,), {f,) in X , F respectively such that p, +. p, s # q,

; fa(p,) -) s, f,(p) -t q. If q E Y then for each cr we have

So u(f,(p,),q) -t 0 and q = s, a contradiction. If s E Y then for each cr we have

a(fa(p),s) 5 P(P,P,) + a(fa(p,),s).

Hence a(f,(p),s) -+ 0 and s = q, a contradiction.

2. UNIFORMLY NORMAL FAMILIES ON HYPERBOLIC MANIFOLDS

In this section we establish some properties of uniformly normal families defined on hyperbolic manifolds. We apply some of these properties to generalize theorems of Brody, of Hahn and of Zaidenberg, and to produce an analogue of a theorem of Aladro and Krantz; these properties are also used to demonstrate that normal maps as defined by various authors in various settings are uniformly normal as singleton sets and that maps in any uniformly normal family (in particular our normal maps)

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

20:

12 0

1 N

ovem

ber

2013

348 J. E. JOSEPH AND M. H. KWACK

are normal maps in the prevailing setting. Furthermore, with these properties we set the stage for those main results which come in Section 3. For r > 0 let D, = { z E C : I z I < r ) and let D: = D, - (0) . Brody has proved the following (see [22], p. 68).

THEOREM A Let X be a relatively compact complex subspace of a complex space Y . If X is not hyperbolically imbedded in Y then there exist sequences {r,), {g,) such that r, > 0 and g, E H(D,,, X ) and a nonconstant g E H(C, Y ) such that r, t ca and g, 4 g on the compact subsets of C.

Remark 2.1 In Theorem A we may assume that r, = n; to see this we first as- sume rl > 1 and that r,+l - r, > 1 by taking a subsequence. If k is a positive integer and k I rl, define fk = gl; if r, < k I r, + I , define fk = gn+l. Then fk E H(Dk, X ) and fk -+ g on compact subsets of C.

DEFINITION 2.2 Let X be a complex space, let Y be a complex space and let F C

H ( X , Y) .

1. A Brody sequence for F is a sequence { f , og,) where f, E F and g, E H(D,,X). 2. A map h E C(C, Y +) is a Brody limit for F i f there is a Brody sequence {h , ) for

F such that h, -t h on the compact subsets of C.

Remark 2.3 If Y is a relatively compact complex subspace of a complex space Z, and X is a complex space, Brody sequences for F c H ( X , Y ) coincide with Zaidenberg's holomorphic curves and Brody limits for F coincide with Zaidenberg's F-limiting mappings [28].

Theorem 2.4 will be used extensively in the sequel.

THEOREM 2.4 Let M be a hyperbolic manifold, and let Y be a complex space. The following statements are equivalent for F c H(M,Y) :

1. F is uniformly normal. 2. For each complex manifold 0, F o H ( 0 , M ) is an evenly continuous subset of

H(Q,Y). 3. F o H(D,M) is an evenly continuous subset of H(D,Y). 4. There is a length function E on Y such that Idf I E 5 1 for each f E F. 5. There is a length function E on Y such that E(h,(O),dh,(O,e)) -+ 0 for each

Brody sequence {h , ) for F . 6. There is a length function E on Y such that E(h,(O), dh,(O, e ) ) -t 0 for each

Brody sequence {h,) for F with a Brody limit.

Proof It is obvious that (1) + (2) + (3) and that ( 5 ) + (6). To show that (3) + (4) we will first show that if (3) holds then for each length function E on Y and compact Q c Y there is a c > 0 such that Idf ( p ) ( 5 c on f -'(Q) for each f E F. The proof that (3) + (4) may then be completed by arguments similar to those in ([21], p. 34). To facilitate the demonstration that each compact Q c Y satisfies the stated property for each length function E on Y we prove a lemma which establishes a property of hyperbolic manifolds which is interesting in its own right.

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

20:

12 0

1 N

ovem

ber

2013

NORMAL M A P S 349

LEMMA 2.5 Let f E H ( M , Y ) where M is a hyperbolic manifold and Y is a complex space with a length function E. Then

Proof Let p E M and v E T p ( M ) such that K M ( p , v ) = 1 and let E > 0. There exist cp E H(D, M ) and r > 0 such that y(0) = p, d(cp(O), re) = v and r < I + 6.

The promised equalities are now immediate. (3) + (4). If a compact Q C Y fails the stated condition for the length function E ,

there are sequences {p,), { f n ) , { v n ) and q E Q, with pn E M , f, E F, v, E Tp,(M), f n ( ~ n ) E Q, KM @n,vn) = 1, f n ( ~ n ) 4 and E(fn(pn), dfn(pn, vn)) > n. It follows that Idfn(pn)l + co and, by Lemma 2.5, some sequence {p,) in H(D,M) satis- fies cp,(O) = p, and (d fn o cpn(0)l --+ co. Let V be a relatively compact neighbor- hood of q hyperbolically imbedded in Y. Since F o H(D,M) is an evenly continu- ous subset of H(D,Y) there is an 0 < r < 1 such that fn o pn(Dr) c V eventually; the sequence of restrictions of { f n o pn) to D,, which we call again { f n o p,}, is uniformly normal and is consequently relatively compact in H(D,,Y). Some sub- sequence of { f n o 9,) converges to h E H(D,, Y ) contradicting Jdfn o pn(0)J -+ co.

(4) + (5). Let E be a length function as promised in (4). If { f n o cp,} is a Brody sequence for F we have

and (5) holds. (4) + (1). From (4) there is a distance function dE on Y such that each f E

F o H(D,M) decreases distances from kD to dE and (1) follows from Proposition 1.6 and 1.10.

(6 ) + (4). If (4) does not hold, then for any length function E on Y there are sequences { f n ) in F, (9,) in H(D,M) such that Jdf , opn(0)l -t m. Utilizing Brody's arguments (see 1221, pp. 68-71) we may obtain a Brody sequence {g,) and Brody limit g for F such that gn --+ g on the compact subsets of C and such that E(gn(0), dgn(0)) = 1 contradicting (6). rn

We point out that equivalence (4) of Theorem 2.4 generalizes the following the- orem of Lehto and Virtanen [24] since all length functions on compact complex spaces are equivalent [21]. Hahn [9] generalized the theorem to f E H(R,Pn(C))

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

20:

12 0

1 N

ovem

ber

2013

350 J . E. JOSEPH AND M. H. KWACK

where R is a bounded homogeneous domain in Cm. It should also be noted that (6) + (4) of Theorem 2.4 can be proved in a different way employing the arguments used in the proof of the generalization of the classical theorem of Lohwater and Pommerenke [23] in Theorem 2.13 below.

THEOREM B A meromorphic function f : D -+ P ~ ( c ) is normal iff Jd f 1 < co. COROLLARY 2.6 Let M be a hyperbolic manifold and let F c H ( M , Y ) be uniformly normal. Then

1. Each Brody sequence for F has a subsequence which converges to a Brody limit for F on the compact subsets of C, and

2. Each Brody limit for F is constant.

Proof From (4) in Theorem 2.4 there is a length function E on Y such that F decreases distances from ku to dE. As for ( I ) , if m is a positive integer and {g,) is a Brody sequence for F, then each g E G = {g, : n 2 m ) decreases distances from kDm to dE, and G is therefore by Proposition 1.10 relatively compact in C(D,, Y + ). For (2), let {g,) be a Brody sequence for F and suppose g is a Brody limit for F such that g, 4 g on the compact subsets of C. If p,q E C and g(p ) , g (q ) E Y , for n large enough we have d~(g,(p) ,g,(q)) 5 k ~ " ( p , q). Since k ~ , ( p , q ) -+ 0 we have g ( p ) = g(q). If g ( p ) = cc the continuity of g and connectedness of C force g(q) = cc since g(C) n Y is at most a singleton.

COROLLARY 2.7 Let M be a hyperbolic manifold, let Y be a complex space and let F c H ( M , Y ) satisfy F ( x ) is relatively compact in Y for each x E M . Then F is uniformly normal iff each Brody limit for F is constant.

Proof The necessity follows from (b) of Corollary 2.6. As for the sufficiency, if F is not uniformly normal, the Brody limit g constructed in (6) + (4) of Theorem 2.4 is nonconstant since Y is a complex space and g(0) E Y .

In the case of P'(c) we have the following result. The proof is omitted as it is readily obtained as a consequence of previous results and Hurwitz's Lemma.

COROLLARY 2.8 Let M be a hyperbolic manifold. The following are equivalent for F c H(M,C) :

1. F is uniformly normal. 2. F is uniformly normal as a subset of H ( M , (c)). 3. I fg is a Brody limit for F and g E H(C,C) then g is constant.

A complex space Y is said to be Brody hyperbolic if every f E H ( C , Y ) is constant. We observe that Y is Brody hyperbolic iff each Brody limit for the identity i : Y -+ Y with values in Y is constant, i.e. if f E H(C,Y) and { f , ) is a sequence such that f, E H(D,,Y) and f, -+ f on the compact subsets of C, then f is constant.

In Corollaries 2.9-2.11 we give characterizations of hyperbolic and hyperbolically imbedded spaces in terms of Brody sequences.

COROLLARY 2.9 A complex space Y is hyperbolic iff there is a length function E on Y such that E ( f , (0), df,(O, e ) ) 4 0 for each sequence {f , ) satisfying f , E H(D,, Y )

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

20:

12 0

1 N

ovem

ber

2013

I NORMAL MAPS 351

and f, -+ g E C(C, Y ' ) on the compact subsets of C. We note that g is necessarily constant.

COROLLARY 2.10 A complex subspace Y of a complex space Z is hyperbolically imbedded in Z iff there is a length function E on Z such that E(f,(O), df,(O, e ) ) -+ 0 for each sequence { f , ) satisfying f , E H(D,,Y) and f , -t g E C(C,Z+) on the com- pact subsets of C; g is necessarily constant.

In view of Corollaries 2.7 and 2.10 we have Corollary 2.11 (compare with Theo- rem 2.3, Chapter I11 in [21]).

COROLLARY 2.11 Let Y be a relatively compact complex subspace of the complex space Z . Then Y is not hyperbolically imbedded in Z iff there is a noncomant g E H(C,Z) and a sequence {g,) such that g, E H(D,,Y) and g, -+ g on the compact subsets of C.

In Corollary 2.12 below we produce an analogue for uniformly normal families of the following result of Aladro and Krantz ([3], Theorem 3.1).

THEOREM C If R is a hyperbolic domain in Cn and M is a complete Hermitian manifold then F c H ( R , M ) is not normal iff there exists a compact Q c R and se- quences {p,) in Q , { f , ) in F , {a,) with a , > 0 and a, -t O f and a sequence (v,) of Euclidean unit vectors in Cn, such that the sequence {g,) in H(C, M ) defined by g,(z) = fn(p, + a,v,z) converges uniformly on the compact subsets of C to a non- constant entire function g.

Corollary 2.12 and Theorem 2.13 also generalize theorems of Lohwater and Pornrnerenke [23]. Theorem 2.13 was proved by Hahn [9] for a normal function when the range space is Pn(C) and the domain is D.

COROLLARY 2.12 Let M be a hyperbolic manifold, let Y be a complex space and let F c H ( M , Y ) . Then F is not uniformly normal iff for each length function E on Y there is a Brody sequence {g,) for F and a Brody limit g for F such that g, -t g and limE(g,(O),dg,(O,e)) > 0. We observe that i fg(C) n Y # 0 then g is not constant.

THEOREM 2.13 Let Y be a complex space and let F c H(D,Y). Then F is not uni- formly normal iff for each length function E on Y there are sequences {f,) in F, {p,) in D, {r,) in ( 0 , ~ ) and {p,) satisfying the following conditions:

(a) rn L 0, rn / ( l - IpnI) -+ 0, (b) pn E H(D,,,,D) is defined by p,(z) = p, + zr, where s, = (1 - /p,l)/r,, (c) fn Opn 4 g E C ( C , Y f ) , and (d) limsup E(f , o p,(z), d f , o p,(z, e ) ) 5 1 for each z E C and

Moreover, g will not be constant in (c) of the necessity if g (C) n Y # 0.

Proof The sufficiency of the conditions is a consequence of Corollary 2.12 and Remark 2.1. For the necessity suppose F is not uniformly normal and let E be a length function on Y . There are sequences {z,) in D, { f , ) in F satisfying

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

20:

12 0

1 N

ovem

ber

2013

352 J. E. JOSEPH AND M. H. KWACK

(d f , (z,)( T m. Define u, > 0 by

Then a , satisfies the following conditions:

1. a; E [1/4,1); (z,l < a,; 1 - ( z , / u , ~ ~ 2 ( 1 - ( ~ , ( ~ ) / 2 and we may assume 2. (1 - lzn/an12)E(fn(zn),dfn(zn,e)) T m.

suppose M, is assumed at p,, and let r, = l /E(f,(p,) , df,(p,,e)). We see that r,/(a, - (p , ( ) + 0, assume that M, T m and that r,/(l - (p , ] ) 0. Let s, = ( 1 - (p , ( ) /r , and define c p , E H(D,,,,D) by cp,(z) = p, + zr,. Now let r > 0 and let s, > r. For z E Dr we have

and the expression on the right hand side of the last inequality tends to 1. Hence { f , o cp,) is evenly continuous on D, with respect to the Euclidean metric on D, and dE on Y . By Proposition 1.10, {f, o cp,) is relatively compact in C ( D , , Y f ) and consequently we may assume that f, o c p , -+ g E C(C, Y + ) . It is immediate that conditions (a), ( b ) and (c) are satisfied by {f,), {p,) , {r,), and {cp,); it is easy to see that

E(fn 0 ~n (O) , d fn 0 v n (0, e ) ) = 1

so (d) holds.

If M is a homogeneous hyperbolic manifold we represent its space of automor- phisms by A ( M ) . Theorem 2.14 is a characterization theorem for uniformly normal families on such manifolds and Corollary 2.15 gives an indication of why we adopted the terminology "uniformly normal families".

THEOREM 2.14 The following statements are equivalent for F c H ( M , Y ) when M is a homogeneous hyperbolic manifold and Y is a complex space:

1. F is uniformly normal. 2. F o A ( M ) is an evenly continuous subset of H ( M , Y ) . 3. F o A ( M ) is relatively compact in C (M,Y+) . 4. F o H ( M , M ) is relatively compact in C ( M , Y f ) .

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

20:

12 0

1 N

ovem

ber

2013

NORMAL M A P S 353

Proof (1) + (2). This is clear from Definition 1.2 and Proposition 1.1. (2) + (3). We need show that F o A ( M ) is evenly continuous from M to Y + .

Suppose x E M , p E Y and {x,) , { f , ) and {p , ) are sequences in M , F and A ( M ) respectively such that X , -+ X , fn o p,(x) --+ oo and f, o p,(x,) -+ p. Let A, E A ( M ) satisfy A,(x) = x,. It follows that fn o c p , o A, is not evenly continuous from x to p so (2) does not hold.

(3) + (1). We show that F o H(D,M) is an evenly continuous subset of H ( M , Y ) and use equivalence (3) of Theorem 2.4. Suppose F o H(D, M ) is not evenly contin- uous from 0 € D to p E Y. There are sequences {z,), { f, o p,) in D, F o H(D, M ) respectively and a neighborhood U of p in Y such that zn + 0, f, o yo,(O) -+ p and f, o p,(z,) $ U is satisfied. Let a E M and let {A,) be a sequence in A ( M ) such that /\,(a) = cp,(O). Then f, o &(a) -+ p while fn o A,(~;~(cp,(z,)) $! U . Since each f E A ( M ) preserves hyperbolic distances and each f E H ( D , M ) is distance decreas- ing with respect to k o and ku we see that Ai l (pn(zn) ) -+ a and this contradicts our hypothesis (3).

The proof of the theorem may be completed by observing that the equivalence of (4) to the other statements follows from the equivalence of (1) and (3), Propositions 1.4 and 1.5, and the inclusion F o A ( M ) c F o H ( M , M ) .

COROLLARY 2.15 Let M be a homogeneous hyperbolic manifold, let Y be a complex space and let F c H ( M , Y ) satisfy F = F o A ( M ) . Then

1. F is uniformly normal iff F is relatively compact in C ( M , Y + ) ; 2. i f M = D and Y = C, F is uniformly normal iff F is normal in the sense of Wu

~ 7 1 .

Proof Assertion (1) is an easy consequence of Proposition 1.4 and (3) of Theo- rem 2.14. Assertion (2) can be seen to be a product of (1) and Hurwitz's Lemma.

Remark 2.16 Hayman [13] called F c H(D,P1(C)) invariant if F = F o A(D) and called an invariant family uniformly normal if it is normal in the sense of Montel. In view of this and Corollary 2.15 we used the terminology "uniformly normal family".

We close this section with additional examples of families of maps which have been studied by authors in various frameworks. Theorems 2.4 and 2.14 may be em- ployed to verify that these families are all uniformly normal.

EXAMPLE 2.17 I f f E H(D,P'(C)) and A c D is a closed disk and d A denotes the boundary of A, let J ( f ( A ) ) and L ( f (A) ) be respectively the spherical area of f (A) and spherical length o f f ( dA) . Let h > 0 and

F (h ) = {f E H(D,P'(c)) : J ( f ( A ) ) 5 hL( f ( A ) ) for each closed disk A c D).

Hayman ([13], p. 164) showed that F(h) is invariant and normal in the sense of Montel. The family F(h) is therefore uniformly normal by Corollary 2.15.

E~AMFLE 2.18 Let M be a complex manifold, r > 0, and F c H ( M , P1 (C)) be a family of maps such that for each f E F there are points a f , bf , c f E P1 (C)- f ( M ) with ~ ( a f , bf ) x ( c f , bf ) x ( c f , a f ) 2 r where x represents the spherical metric.

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

20:

12 0

1 N

ovem

ber

2013

354 J . E. JOSEPH AND M. H. KWACK

Carathkodory ([5], p. 202) showed in this situation that F o H(D, M ) is normal in the sense of Montel. So F is uniformly normal.

The maps defined in Examples 2.1s2.25 are all normal maps in our sense.

EXAMPLE 2.19 Lehto and Virtanen [24] defined f E H(Q,P~(c ) ) to be normal i f f o A(R) is a normal family in the sense of Montel where R is a bounded homoge- neous domain in C.

EXAMPLE 2.20 Hahn [14] called f E H(R, Y ) normal i f f o H(D, R) is normal in the sense of Wu [27] where R is a bounded domain in Cn and Y is a relatively compact complex subspace of a Hermitian manifold.

EXAMPLE 2.21 Funahashi [7] called f E H(R,Y) normal i f f o A(R) is compact in H(R , Y ) where R is a bounded homogeneous domain in Cn and Y is a complex space.

EXAMPLE 2.22 Cima and Krantz [5] called f E H(R, P1(C)) normal i f Id f (2, v)l 5 cKn(z, v ) for some c > 0 where R is a hyperbolic domain in Cn. They showed that f is normal iSf f o H(D,R) is relatively compact in H(R ,P~(c ) ) .

EXAMPLE 2.23 Krantz ([20], p. 115) defined f E H(R,C) to be a Bloch map if Idf ( p , v ) \ 5 cKn(p, v ) for some c > 0 where R is a hyperbolic domain in Cn.

EXAMPLE 2.24 Aladro and Krantz [3] called f E H(R,Y) normal i f there is a c > 0 such that E ( f (p) ,df (p ,v) ) 5 cKn(p,v) where R is a hyperbolic domain in Cn and Y is a complete complex Hermitian manifold with Hermitian length function E.

EXAMPLE 2.25 Joseph and Kwack [16] have proved that a complex subspace X of a complex space Y is hyperbolically imbedded in Y iff Fx,r is relatively compact in C(D, Y ' ) and hence iff Fx,y n H(D,Y) is a uniformly normal family.

3. UNIFORMLY NORMAL FAMILIES DEFINED ON ARBITRARY COMPLEX SPACES. HIGHER DIMENSIONAL VERSIONS OF CLASSICAL RESULTS OF SCHOTTKY, LAPPAN, AND BOHR FOR UNIFORMLY NORMAL FAMILIES

In this section we apply results in Sections 1 and 2 to establish properties of uni- formly normal families on arbitrary complex spaces and we generalize theorems of Schottky, Hayman and Lappan by replacing the bounded domains in C by arbitrary complex spaces and the maps under consideration by uniformly normal families. A Lindelof principle shown by Cima and Krantz to be satisfied by normal maps from a bounded domain in Cn to P ~ ( c ) is generalized by replacing P1(C) by an arbitrary complex space and normal maps by uniformly normal families.

The following proposition on Brody sequences and limits will be useful.

P R O P ~ S I T I ~ N 3.1 Let X , Y be complex spaces and let F C H ( X , Y ) . Then

1. {g,) is a Brody sequence for F iff {g,} is a Brody sequence for F o H(D,X) , and

2. g is a Brody limit for F iff g is a Brody limit for F o H(D,X) .

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

20:

12 0

1 N

ovem

ber

2013

1 NORMAL MAPS- 355

Proof Conclusion (2) is an easy consequence of conclusion (1). As for ( I ) , if g, = f, o c p , where f , E F and cp, E H(D,,X), then g, = fn 0 c p , o m, 0 m;' where m, E H(D,D,) is multiplication by n. Then f , o c p , o m, E F o H(D,X) and m i 1 E H(D,,D), so { g , ) is a Brody sequence for F o H ( D , X ) if {g,) is a Brody sequence for F. On the other hand, if g, = h, o c p , where h, E F o H(D,X) and cp , E H(D,,D), then g, = f , 0 a, 0 yo, where fn E F and a, 0 cp, E H(Dn, X ) . So each Brody sequence for F o H(D,X) is a Brody sequence for F.

Theorem 3.2 is a consequence of Definition 1.2, Propositions 1.6, 3.1 and Theo- rem 2.4.

THEOREM 3.2 If X , Y are complex spaces the following statements are equivalent for F c H ( X , Y ) :

1. F is uniformly normal. 2. F o H ( M , X ) is an evenly continuous subset of H ( M , Y ) for each complex man-

ifoki M. 3. F o H(D,X) is an evenly continuous subset of H(D,Y). 4. There is a length function E on Y such that (dg I E 5 1 for each g E F o H(D, X ) . 5. There is a length function E on Y such that E(h,(O),dh,(O,e)) -+ 0 for each

Brody sequence {h,) for F. 6. There is a length function E on Y such that E(h,(O),dh,(O,e)) -+ 0 for each

Brody sequence {h,} for F with a Brody limit.

Propositions 1.6, 3.1 and Corollary 2.6 lead to the following result.

THEOREM 3.3 Let X be a complex space and F c H ( X , Y ) be uniformly normal. Then

1. Each Brody sequence for F has a subsequence which converges to a Brody limit for F on the compact subsets of C.

2. Each Brody limit for F is constant.

Theorem 3.4 follows from Propositions 1.6, 3.1 and Corollary 2.7.

THEOREM 3.4 Let X , Y be complex spaces and let F c H ( X , Y ) satisfy F(x) is relative& compact in Y for each x E X . Then F is uniformly normal iff each Brody limit for F is constant.

For uniformly normal families of functions we have the following theorem.

THEOREM 3.5 The following statements are equivalent for a complex space X and F c H ( X , C):

1. F is a uniformly normal fami&. 2. F is a ungormly normal family as a subset of H(X,P1(C)). 3. I f g E H(C,C) is a Brody limit for F then g is constant.

Proof See Corollary 2.8.

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

20:

12 0

1 N

ovem

ber

2013

356 J. E. JOSEPH AND M. H. KWACK

Theorem 3.6 is a generalization of a theorem of Lohwater and Pommerenke to uniformly normal families on arbitrary complex spaces (see Corollary 2.12 and The- orem 2.13).

THEOREM 3.6 Let X ,Y be complex spaces and let F c H ( X , Y ) . Then F is not uniformly normal iff for each length function E on Y there is a Brody sequence {g,) for F and a Brody limit g for F such that g, -+ g, lim sup E(g, (z) , dg, ( z , e ) ) 5 1 for each z E C and E(g, (0), dg, (0, e ) ) = 1.

Hayrnan ([13], p. 165) proved the following strong version of Schottky's Theorem.

THEOREM D Let F c H(D,C) be a normal invariant family. Then there is a c > 0 depending only on F such that

for each f E F, 0 5 r < 1, where p = maxi 1, 1 f ( 0 ) ) ) .

If X is a complex space, f E H ( X , C ) and x E X we denote max{l, ( f ( x ) ( ) by p( f , x ) . It is obvious that the conclusion of Theorem D can be replaced by the existence of c > 0 depending only on F such that

for each f E F and z E D. Zaidenberg [28] extended Hayrnan's result to uniformly normal families of func-

tions defined on complex manifolds. This generalization was effected by use of the Kobayashi-Royden-Green metric which was introduced in [8] and studied further in [29]. We utilize parts of Hayman's proof of Theorem D to extend Zaidenberg's result to uniformly normal families of functions defined on complex spaces with more elementary proofs.

THEOREM 3.7 Let X be a complex space. The following statements are equivalent for F c H(X ,C) :

1. F is uniformly normal. 2. There is a c > 0 such that all f E F, x,y E X satisfy the inequality

3. There is a c > 1 such that all f E F, x, y E X satisfy the inequality

4. There is a c > 1 such that all f E F , x E X and 0 f Q c X satisfy the inequal- ity

r 1

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

20:

12 0

1 N

ovem

ber

2013

NORMAL M A P S 357

5. There is a c > 1 such that all f E F , x, y E X satisjj the inequality

6. There is a c > 1 such that all f E F, cp E H(D,X) , x,y E D satisfy the inequality

Proof (1) =. (2). Clearly F o H ( D , X ) is uniformly normal and invariant, and is consequently normal in the sense of Monte1 by Corollary 2.15. Hayman ([13], p. 165) has shown that there is a c > 0 such that for all g E F o H ( D , X ) ,

where PO = max{l , Jg(0)l). For z E D define $, E A(D) by $,(w) = (w + z ) / ( 1 + Zw). Then for each g E F o H ( D , X ) we have

where pz = max{l , lg(z)l). Let x,y E X , let f E F and let E > 0. There exist an inte- ger j > 1, cpl,cpz,. . .,Yj E H ( D , X ) and al,. . .,aj E (0 , l ) satisfying cpl(0) = y, cpi(ai) =pi+l(O) for i = I , . . . , j - 1 and p j ( a i ) = x . We may assume that If(x)l > 1, that gi([O, ail) C Or gi([O, ail) C C - D, where gi = f o cpi, and that xi kD(O, a i ) < k x ( ~ , y ) + ~ / 2 . Let I = {i :g i ( [O,a i] )CC-D) . For each i~ I ,

Igi' (2 ) 1 <- 2 for z E [O,ai]; Igi(z)l(logki(z)l+ C) - 1 - 121'

hence for each i E I we have

If 1 E I then

log [loglf + '1 2kx (x , y ) + r. loglf (Y)I + c

If 1 @ I let a be the smallest element of I . Then lg,(0)1 = 1 and

log [ log I f + ] < Zkx ( x , y ) + E. loglga(0)l + c -

The proof is easy to complete. (2) + (3). Replace c with logc in the conclusion of (2). (3 ) + (4). If 0 f Q C X and x E X , let {y,) be a sequence in Q such that

dx(x,y , ) -, d x ( x , Q). The conclusion of (4) is then evident. (4) + (5). Obvious. (5) + (6). I f f E F , cp E H ( D , X ) and x,y E D we have, from (5 ) ,

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

20:

12 0

1 N

ovem

ber

2013

358 J . E. JOSEPH AND M. H. KWACK

(6) + (1). In view of Proposition 1.6 and the homogeneity of D, it suffices to show that if { f , ) and (9 , ) are sequences in F and H ( D , X ) , respectively, some sub- sequence of {f, o 9,) converges to g E C ( V , Y + ) on some neighborhood V of 0. If for each compact K c Y and neighborhood V of 0, f, o p,(V) n K = 0 eventually, we finish. Otherwise we may assume a sequence {z,) in Dl12 such that { f , o p,(z,)) is bounded. There is a c > 1 such that, for each z E D1p,

and { f , o p,) is uniformly bounded on

In Corollary 3.8 we apply equivalence (2) in Theorem 3.7 to extend the follow- ing lemma of Bohr to holomorphic functions defined on complex spaces (see [13], p. 50).

LEMMA E Suppose that w = f ( z ) is regular in Izl 5 1, satisfies f (0) = 0 and

Then f ( z ) assumes in lzl < 1 all values on some circle Iwl = r, where r > A and A is a positive absolute constant.

COROLLARY 3.8 Let X be a complex space, and let B c X , f E H ( X , C ) satisfj ( 1 ) B is bounded with respect to the pseudo-metric k x , (2) sup,,, ( f ( x ) ( > 1 and (3) 0 E f (B) . Then there is an r > 0 independent o f f such that

Proof We note that H ( X , C - {O,l)) is a uniformly normal family in H ( X , C ) . By Theorem 3.7(2), there is a c > 0 such that

for all p,q E B , a E H(X,C - {0,1)) such that la(q)( _< 1; let

Suppose

and define cp E H ( X , C - {0,1)) by cp(p) = (f ( p ) - wl)/(w2 - wl). There is a q E B such that f ( q ) = 0; hence (cp(q)J L. 1 and, for each p E B , [cp(p)( < A and

This is a contradiction.

Remark 3.9 Equivalences (1) + (3) + (4) 3 (1) of Theorem 3.7 are due to Zaidenberg [28] for uniformly normal families of functions defined on complex manifolds.

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

20:

12 0

1 N

ovem

ber

2013

NORMAL MAPS 359

In Theorem 3.11 we provide a generalization of the following Five-Point Theorem of Lappan [22].

THEOREM F Let A be a subset of P'(c) with at least 5 elements. Then

is normal iff

We extend this theorem of Lappan to uniformly normal families from arbitrary complex spaces to Pn(C) by first extending the theorem to such families defined on hyperbolic manifolds and then using equivalence (2) of Propositions 1.6 and 3.1 (also see [9, 11, 121). We say that g E H(Cm,Pn(C)) is degenerate if g(Cm) c w for some hyperplane w in Pn(C).

THEOREM 3.10 Let M be a hyperbolic manifold, let u be a collection of at least 2n + 3 hyperplanes in general position in Pn(C) and let A = U , w. Then F c H ( M , Pn(C)) is uniformly normal iff the following two conditions hold:

1. sup{ldf(p)l : P E U,f - l ( ~ ) ) < and 2. Each degenerate Brody limit for F is constant.

Proof The necessity of the two conditions is obvious from equivalences (4) and (6) of Theorem 2.4 since all length functions on Pn(C) are equivalent. To establish the sufficiency we will show that if condition (2) holds and F is not uniformly nor- mal then condition (1) fails. To effect this demonstration we use some additional terminology and the following lemma of Hahn [9] which generalizes a well-known result in Nevanlinna Theory for meromorphic functions (see 1141, p. 231, Corollary 3). If a is a hyperplane in Pn(C) let T, be a nonzero linear functional on C n f l for which a is the kernel. If a is a collection of hyperplanes in general position in Pn(C) we say that a nondegenerate g E H(Cm,Pn(C)) is totaUy ramifed over u if for each a E a, (T, o g)'(C) = 0 whenever T, og(<) = 0.

LEMMA J Let u = {a1,. . . ,aq) be any collection of hyperplanes in general position in Pn(C). I f g E H(Cm, Pn(C)) is total& rami@d over a , then q 5 2n + 2.

Now, to complete the proof. Since P ( C ) is compact and F is not uniformly normal there is a nonconstant Brody limit g E H(C,Pn(C)) for F by equivalence (6) of Theorem 2.4. Let i f k ) , {$k) be sequences such that fk E F, $k E H(Dk,M), and fk o $k t g. From (2), g is nondegenerate and itom Lemma J, g is not totally ramified over ( a ) for some a E a. Choose homogeneous coordinates [" w, . . . ," w] of Pn(C) so that a is defined by Ow = 0. If gk = fk OT+!J~ we represent g and gk by the homogeneous coordinates ["g, . . . , "g ] and [Ogk, . . . , "gk], respectively, where $g and $gk are holomorphic and "gk -+ "g. The equation "g(z) = 0 has a solution zo E C for which "gi(zo) # 0 and hence, if E is a length function on Pn(C), we have E(g(zo),dg(zo,e)) = a # 0. By Hurwitz's Lemma there is a sequence { z k ) in

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

20:

12 0

1 N

ovem

ber

2013

360 J. E. JOSEPH AND M. H. KWACK

C such that Zk 4 20, Ogk (zk) = 0 and E(gk (zk), dgk ( Z k , e ) ) -+ QI. Let pk = fi (zk) . Then

so Jd fk (pk )J 4 oo. Since pk E f ; ' ( ~ ) c f;l(A), condition (1) does not hold. . We are now in a position to prove the following generalization of the theorem of

Lappan.

THEOREM 3.11 Let X be a complex space, let c be a collection of at least 2n + 3 hyperplanes in general position in Pn(C) and let A = U,T. Then F c H(X,Pn(C)) is uniformly normal iff the following two conditions hold:

1. sup{ ldf 0 cp(0)I : cp E H(D, X ) , ~ ( 0 ) E U F f --l(A)} < oo, and 2. each degenerate Brody limit for F is constant.

Proof From Theorem 3.10 and Proposition 1.6 we have that F is uniformly nor- mal iff the following two conditions hold:

(1") sup{Idg(p)l : g E FoH(D,X) , P E Ug-'(A)} < m, and (2") Each degenerate Brody limit function for F o H(D, X ) is constant.

Conditions ( lo) and (2") are equivalent to conditions (1) and (2) respectively in view of Proposition 3.1 and the following set equality:

The next two corollaries are now readily available.

COROLLARY 3.12 Let X be a complex space. Then F c H ( X , P'(c)) is uniformly normal iff

where A c pl(C) has more than 4 elements (2 jinite elements i f F c H ( X , C)) .

COROLLARY 3.13 Let M be a hyperbolic manifold. Then F c H ( M , P ~ ( C ) ) is uni- formly normal iff

Idf(p)l : PE U f - ' ( 4 F

where A c P'(c) has more than 4 elements (2jinite elements i f F c H ( M , C)).

We close with a theorem on the boundary behavior of uniformly normal families on bounded domains in Cn with values in a complex space, and a final remark. In [5] Cima and Krantz introduced the concept of hypoadmissible limit to study

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

20:

12 0

1 N

ovem

ber

2013

NORMAL MAPS 361

the boundary behavior of normal maps on higher dimensional domains. One of the results which they established was the following:

THEOREM G I f R is a bounded domain in Cn with c2 boundary and

is normal with radial limit q E pl(C) at p E dR then f has hypoadmissible limit q at p.

We extend the notions of radial and hypoadmissible limits to uniformly normal families from bounded domains in Cn to complex spaces in Definition 3.14 and extend Theorem G to such families in the form of Theorem 3.15. We omit the proof of our theorem since arguments similar to those of Cima and Krantz may be used in view of equivalence (4) of Theorem 2.4 which implies that the members of such families decrease distances with respect to some metric on the range.

D E ~ N I T I O N 3.14 Let R be a bounded domain in Cn, let Y be a complex space and let F c H(R ,Y) .

1. F has a radial limit q E Y at p E dR i f f,(p - E,,v,) -+ q for all sequences { f , ) in F and {E,) in (0 , l ) such that E, + 0 where vp is the unit outward normal at p.

2. R c R is a hypoadmissible approach region at p E dR if R is an open connected subregion of R satisfying the following three properties: (a) There is an a > 0 such that I',(p) c R where

3. F has hypoadmissible limit q at p if f,(p,) + q for each sequence { f , ) in F, hypoadmissible approach region R c R at p and sequence {p,} in R.

THEOREM 3.15 Let R c Cn be a bounded domain with c2 boundary, let Y be a complex space and let F c H ( R , Y ) be uniform& normal. I f F has radial limit q E Y at p E dR then F has hypoadmissible limit q at p.

Remark 3.16 For those results in this paper where the family F is assumed to satisfy the condition that F ( x ) is relatively compact in the range space Y for each x, C may be replaced by H and Y + by Y . In all cases where the additional assumption is made that all members of F map into a fixed compact subset of the range space Y, C may be replaced by H and Y f by Y .

References [I] M. Abate, A characterization of hyperbolic manifolds, Proc. Amel: Math. Soc. 117 (1993), 789-793. [2] G. Aladro, Applications of the Kobayashi metric to normal functions of several complex variables,

Utilitas Math. 31 (1987), 13-24.

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

20:

12 0

1 N

ovem

ber

2013

362 J. E. JOSEPH AND M. H. KWACK

(31 G. Aladro and S. G. Krantz, A criterion for normality in Cn, J. Math. Anal. and Appl. 161 (1991), 1-8.

[4] C. Carathbdory, Theory of Functions, vol. 11. Chelsea, NY, 1954. [S] J. A. Cima and S. G. Krantz, The Lindelof principle and normal functions of several complex

variables, Duke Math. J. 50 (1983), 303-328. [6] E. E Collingwood and A. J. Lohwater, The Theory of Cluster Sets, Cambridge University Press,

London, 1966. [T K. Funahashi, Normal holomorphic mappings and classical theorems of function theory, Nagoya

Math. J. 94 (1984), 89c104. [8] M. L. Green, The hyperbolicity of the complement of 2n + 1 hyperplanes in general position in Pn,

and related results, Proc. h e r : Math. Soc. 66 (1977), 109-113. [9] K. T Hahn, Higher dimensional generalizations of some classical theorems on normal meromorphic

functions, Complex Variables 6 (1986), 10P-121. [lo] K. T. Hahn, Non-tangential limit theorems for normal mappings, Pac. J. Math. 135 (1988), 57-64. [ l l ] K. T. Hahn, Boundary behavior of normal and nonnormal holomorphic mappings, Proc. KIT Math.

Worhhop, Analysis and Geometry, 1987, KIT Math. Research Center, Taejon, Korea. [I21 K. T. Hahn, Hyperbolicity of the complement of closed subsets in a compact Hermitian manifold,

Complex Anal. and Appl. '87, Sofia 1989,211-218. [13] W. K. Hayman, Meromorphic Functions, Oxford University Press, Oxford, 1964. [14] E. Hille, Analytic Function Theory, vol. 11, Ginn, Lexington, MA, 1%2. [IS] P. Jarvi, An extension theorem for normal functions in several variables, Proc. A4.9 103 (1988),

1171-1174. [16] J. E. Joseph and M. H. Kwack, Hyperbolic imbedding and spaces of continuous extensions of holo-

morphic maps, J. Geom. Analysis 4, 3 (1994), 361-378. [ IT J. L. Kelley, General Topology, Van Nostrand, Princeton, NJ, 1955. [I81 P. Kieman, Hyperbolically imbedded spaces and the big Picard theorem, Math. Ann. 204 (1973),

203-209. [19] S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings, Marcel Dekker, New York, 1970. [20] S. G. Krantz, Geometric Analysis and Function Spaces, CBMS, h e r : Math Soc. 81, Providence, RI,

1993. [21] S. Lang, Introduction to Complex HjJperbolic Spaces, Springer-Verlag, NY, 1987. [22] P. Lappan, A criterion for a meromorphic function to be normal, Comment. Math. Helvetici 49

(1974), 492-495. [23] A. J. Lohwater and Ch. Pommerenke, On normal meromorphic functions, Ann. Acad. Sci. Fenn.

Sex A1 550 (1973). [24] 0. Lehto and K. I. Virtanen, Boundary behaviour and normal meromorphic functions, Acta Math.

97 (1957), 47-65. [ Z ] K. Noshiro, Contributions to the theory of meromorphic functions in the unit circle, J. Fac. Sci.

Hokkaido Univ. 7 (1938), 14%159. [26] H. Royden, Remarks on the Kobayashi metric, Proc. Maryland Conference on Several Complex Vari-

ables, Lecrure Notes 185, Springer-Verlag, Berlin, 1971. [27l H. Wu, Normal families of holomorphic mapping, Acta Math. 119 (1%7), 193-233. [28] M. G. Zaidenberg, Schottky-Landau growth estimates for s-normal families of holomorphic map-

pings, Math. Ann 293 (1992), 123-141. [29] M. G. Zaidenberg, Picard's theorem and hyperbolicity, Siberian Math. J. 24 (1983), 858-867.

Dow

nloa

ded

by [

Mos

kow

Sta

te U

niv

Bib

liote

] at

20:

12 0

1 N

ovem

ber

2013