NORMAL FAMILIES OF MEROMORPHIC MAPPINGS SHARING HYPERSURFACES

In this paper, we prove some normality criteria for families of meromorphic mappings of a domain D ⊂ C m into the complex projective space CP n under a condition on the inverse images (ignoring counting multiplicities) of moving hypersurfaces

NORMAL FAMILIES OF MEROMORPHIC MAPPINGS

SHARING HYPERSURFACES

Nguyen Thi Thu Hang[1] and Tran Van Tan[2]

Department of Mathematics, Hanoi National University of Education

136- Xuan Thuy street, Cau Giay, Hanoi, Vietnam

[1]thuhanghp2003@gmail.com, [2]tranvantanhn@yahoo.com

Abstract

In this paper, we prove some normality criteria for families of mero-morphic mappings of a domain D ⊂ Cm into the complex projective

space CPnunder a condition on the inverse images (ignoring counting

multiplicities) of moving hypersurfaces

Keywords: Normal family, Meromorphic mappings, Moving

hypersur-faces, Nevanlinna theory

Mathematics Subject Classification 2000: 32A10, 32C10, 32H20

1 Introduction

The Little Picard Theorem states that if a meromorphic function on the com-plex plane C omits three distinct points in C, then it is a constant function; and the classical result of Montel says that the family F of meromorphic functions on a domain D ⊂ C is normal if there are three distinct points

a, b, c ∈ C such that each element of F omits each of a, b, and c in D The Little Picard Theorem was generalized to the case of entire curves in the complement of 2n + 1 hyperplanes in general position in CPn by Fujimomto [5], and to the case of entire curves in the complement of 2n+1 hypersurfaces

in general position in CPn by Eremenko [4] According to Bloch’s principle,

to every Picard-type theorem there should belong a corresponding normality criterion The normality result corresponding to the aforementioned Picard-type theorems was proved by Tu [12], and Tu-Li [13] In this paper, we examine this problem for the case where the mappings of the family can meet the hyperplanes (and hypersurfaces)

Let f be a meromorphic mapping of a domain D in Cm into CPn Then for each a ∈ D, f has a reduced representation ef = (f0, · · · , fn) on a neigh-borhood U of a in D which means that each fi is a holomorphic function on

U and f (z) = (f0(z) : · · · : fn(z)) outside the analytic set (of all points of indetermination of f ) I(f ) := {z : f0(z) = · · · = fn(z) = 0} of codimension

≥ 2

In 1974, Fujimoto [6] introduced the notion of a meromorphically normal family into the complex projective space

A sequence {fk}∞

k=1of meromorphic mappings of a domain D in Cminto CPn

is said to converge meromorphically on D to a meromorphic mapping f of D into CPn if and only if, for any z ∈ D, each fk has a reduced representation e

fk = (fk0, · · · , fkn) on some fixed neighborhood U of z such that {fki}∞

k=1

converges uniformly on compact subsets of U to a holomorphic function fi (0 ≤ i ≤ n) on U with the property that (f0, · · · , fn) is a representation of

f in U

A family F of meromorphic mappings of a domain D in Cminto CPnis said to

be meromorphically normal on D if any sequence in F has a meromorphically convergent subsequence on D

Denote by HD the ring of all holomorphic functions on D Let Q be a homogeneous polynomial in HD[x0, , xn] of degree d ≥ 1 Denote by Q(z) the homogeneous polynomial over C obtained by substituting a specific point

z ∈ D into the coefficients of Q We define a moving hypersurface in CPn

to be any homogeneous polynomial Q ∈ HD[x0, , xn] such that the coeffi-cients of Q have no common zero point We say that moving hypersurfaces {Qj}qj=1 (q ≥ t + 1) in CPn are in t−subgeneral position with respect to a subset X ⊂ CPn, if there exists z ∈ D such that for any 1 ≤ j0 < · · · < jt≤ q the system of equations

Qji(z)(w0, , wn) = 0

0 ≤ i ≤ t has no solution (w0, , wn) 6= (0, , 0) satisfying (w0 : · · · : wn) ∈ X

Let Q be a moving hypersurface in CPn, and let f be a meromorphic map-ping of D into CPn For z ∈ D, take ˜f = (f0, , fn) is a reduced representa-tion of f in a neighborhood of U of z The divisor ν(f,Q)(z) := νQ( ˜f )(z)(z ∈ U )

is determined independently of a choice of reduced representations, and hence

is well-defined on the totality of D Here, νQ( ˜f ) is the zero divisor of holo-morphic function Q( ˜f )

In [6], Fujimoto obtained the following interesting result

Theorem F Let F be a family of meromorphic mappings of a domain

D ⊂ Cm into CPn and let {Hj}(2n+1)j=1 be hyperplanes in CPn in general position such that for each f ∈ F , f (D) 6⊂ Hj (j = 1, , 2n + 1) and for any fixed compact subset K of D, the 2(m − 1)-dimensional Lebesgue areas

of f−1(Hj) ∩ K (j = 1, , 2n + 1) inclusive of multiplicities for all f in F are bounded above by a fixed constant Then F is a meromorphically normal family on D

We refer the readers to [3,9,11,12,13], for extensions of the above theorem for both cases of fixed and moving hypersurfaces I would like to remark that

in all of these results, the multiplicity of intersections are taken into account

In this paper, we obtain the following results, where multiplicity of inter-sections are disregarded

Theorem 1.1 Let X ⊂ CPn be a projective variety Let Q1, , Q2t+1 be moving hypesurfaces in CPn in t-subgeneral position with respect to X Let

F be a family of meromorphic mappings f of a domain D ⊂ Cm into X, such that Qj(f ) 6≡ 0, for all j ∈ {1, , 2t + 1} Assume that

a) f−1(Qj) = g−1(Qj) (as sets) for all f, g in F , and for all j ∈ {1, , 2t+ 1},

b) dim(∩2t+1j=1 f−1(Qi)) ≤ m − 2 for f ∈ F

Then F is a meromorphically normal family on D

For each moving hyperplane H in CPndefined by the homogeneous poly-nomial H := a0x0 + · · · + anxn ∈ HD[x0, , xn] (the coefficients a0, , an have no common zero point), we define a holomorphic map H∗ of D into

CPn with the reduced representation (a0 : · · · : an) Let 2t + 1(t ≥ n) moving hyperplanes Hj := aj0x0+ · · · + ajnxn ∈ HD[x0, , xn] (j = 1, , 2t + 1) Define

D(H1, , H2t+1) := Y

L⊂{1, ,2t+1},#L=t+1

{j 0 , ,j n }⊂L

det(aj`i)0≤i,`≤n

)

The moving hyperplanes {Hj} are said to be in pointwise t−subgeneral posi-tion if for every z ∈ D, the fixed hyperplanes {Hj(z)} in CPnin t−subgeneral position It is clear that H1, , H2t+1are in pointwise t−subgeneral position

in CPn if and only if D(H1, , H2t+1)(z) > 0, for all z ∈ D

For the case of hyperplanes, we obtain the following result

Theorem 1.2 Let F be a family of meromorphic mappings of a domain

D ⊂ Cm into CPn For each f in F , we consider 2t + 1 moving hyperplanes

H1f, · · · , H(2t+1)f in CPn such that {Hjf∗ : f ∈ F } (j = 1, , 2t + 1) are normal families and there exists a positive constant δ0 satisfying

D(H1f, , H(2t+1)f)(z) > δ0, for all z ∈ D, f ∈ F

Let m1, , m2t+1 be positive integers and may be ∞ such thatP2t+1

j=1

1

m j < 1 Assume that Hjf(f ) 6≡ 0 for all j ∈ {1, , 2t + 1}, f ∈ F , and two following conditions are satisfied:

a) {z : 1 ≤ ν(f,Hjf)(z) ≤ mj} = {z : 1 ≤ ν(g,Hjg)(z) ≤ mj} for all f, g in

F , and for all j ∈ {1, , 2t + 1},

b) I(f ) ⊂ ∪2t+1j=1 {z : 1 ≤ ν(f,Hjf)(z) ≤ mj}, and Hjf(f ) 6≡ 0, for all

j ∈ {1, , 2t + 1} and f ∈ F , where I(f ) is the set of all points of indeter-mination of f

Then F is a meromorphically normal family on D

Acknowledgements: This research is funded by Vietnam National Foun-dation for Science and Technology Development (NAFOSTED) The second named author was partially supported by Vietnam Institute for Advanced Study in Mathematics, and by the Abdus Salam International Centre for Theoretical Physics, Italy We thank Si Duc Quang for helpful discussions

2 Notations

Let ν be a nonnegative divisor on C For each positive integer (or +∞) p,

we define the counting function of ν (where multiplicities are truncated by p) by

N[p](r, ν) :=

r

Z

1

n[p]ν (t)

t dt (1 < r < +∞)

where n[p]ν (t) =P

|z|≤tmin{ν(z), p}

For a holomorphic mapping f of C into CPn, and a homogeneous polynomial

Q in CPn, with Q( ˜f ) 6≡ 0, we define N(f,Q)[p] (r) := N[p](r, ν(f,Q)) We simply write N(f,Q)(r) for N(f,Q)[+∞](r)

Let f be a holomorphic mapping of C into CPn For arbitrary fixed homo-geneous coordinates (w0 : · · · : wn) of CPn, we take a reduced representation e

f = (f0, · · · , fn) of f Set kf k = max{|f0|, , |fn|}

The characteristic function of f is defined by

Tf(r) := 1

2π

2π

Z

0

logkf (reiθ)kdθ − 1

2π

2π

Z

0

logkf (eiθ)kdθ, 1 < r < +∞

We state the First and the Second Main Theorems in Nevanlinna theory: First Main Theorem Let f be a holomorphic mapping of C into CPn and

Q be a homogeneous polynomial in C[x0, · · · , xn] of degree d ≥ 1, such that Q(f ) 6≡ 0 Then

N(f,Q)(r) 6 d · Tf(r) + O(1) for all r > 1

Second Main Theorem Let f be a linearly nondegenerate holomorphic mapping of C into CPn and H1, , Hq be fixed hyperplanes in CPn in

N −subgeneral position (q > N ≥ n) Then,

(q − 2N + n − 1)Tf(r) 6

q

X

j=1

N(f,H[n]

j )(r) + o Tf(r)

for all r except for a subset E of (1, +∞) of finite Lebesgue measure

3 Proof of our results

In order to prove Theorems 1.1-1.2, we need some preparations

Definition 3.1 ([13], Definition 4.4) Let {νi}∞

i=1 be a sequence of non-negative divisors on a domain D in Cm It is said to converge to a non-negative divisor ν on D if and only if any a ∈ D has a neighborhood U such that there exist nonzero holomorphic functions h and hi on U with νi = νhi and ν = νh on U such that {hi}∞

i=1 converges to h uniformly on compact subsets of U

Let S be an analytic set in D of codimension ≥ 2 By Thullen-Remmert-Stein’s theorem [14], any non-negative divisor ν on D \ S can be uniquely extended to a divisor bν on D Moreover, we have

Lemma 3.1 ([6], page 26, (2.9)) If a sequence {νk}∞

k=1 of non-negative divisors on D \ S converges to ν on D \ S, then {νbk} converges to ν on D.b Lemma 3.2 ([6], Proposition 3.5) Let {fi} be a sequence of meromorphic mappings of a domain D in Cm into CPn and let E be a thin analytic subset

of D Suppose that {fi} meromorphically converges on D \ E to a meromor-phic mapping f of D \ E into CPn If there exists a hyperplane H in CPn

such that f (D \ E) 6⊂ H and {ν(fi,H)} is a convergent sequence of divisors on

D, then {fi} is meromorphically convergent on D

Similarly to Corollary 3.1 in [6], we give the following corollary:

Corollary 3.1 Let S be an analytic set of codimension ≥ 2 in a domain

D ⊂ Cm Let {fk}∞

k=1 be a sequence of meromorphic mappings of D into

CPn Assume that {fk|D\S} meromorphically converges to a meromorphic mapping f of D \ S, then {fk} is meromorphically convergent on D

Proof Let H be a hyperplane in CPn such that f (D \ S) 6⊂ H It is clear that {ν(fk,H)} converges on D \ S Then, by Lemma 3.1, {ν(fk,H)} converges

on D Therefore, by Lemma 3.2, {fk} meromorphically converges on D Lemma 3.3 ([1], Theorem 3.1) Let F be a family of holomorphic mappings

of a domain D in Cm into CPn The family F is not normal on D if and only

if there exist sequences {pj} ⊂ D with {pj} → p0 ∈ D, {fj} ⊂ F , {ρj} ⊂ R with ρj > 0 and {ρj} → 0, and Euclidean unit vectors {uj} ⊂ Cm, such that gj(ξ) := fj(pj + ρjujξ), where ξ ∈ C satisfies pj + ρjujξ ∈ D, converges uniformly on compact subsets of C to a nonconstant holomorphic mapping g

of C into CPn

Lemma 3.4 ([4], Theorem 1) Let X be a closed subset of CPn (with respect

to the usual topology of a real manifold of dimension 2n) and let D1, , D2`+1

be (fixed) hypersurfaces in CPn, in `−subgeneral position with respect to X Then, every holomorphic mapping f of C into X \ (∪2`+1j=1 Dj) is constant Proof of theorem 1.1 Assume that X is defined by homogeneous poly-nomials Q2t+2, , Qs in C[x0, , xn] By replacing Qi by Qdj

j where dj is

a suitable positive integer, we may assume that Qj (j = 1, , s) have the same degree d

Set

Td:=(i0, , in) ∈ Nn+10 : i0+ · · · + in= d

Assume that

Qj = X

I∈Td

ajIxI (j = 1, , s)

where ajI ∈ HD for all I ∈ Td, j ∈ {1, , 2t + 1}, ajI ∈ C for all I ∈ Td, j ∈ {2t + 2, , s}, xI = xi0

0 · · · xi n

n for x = (x0, , xn) and I = (i0, , in) Let T = ( , tkI, ) (k ∈ {1, , s}, I ∈ Td) be a family of variables Set

e

Qj = X

I∈T d

tjIxI ∈ Z[T, x], j = 1, , s

For each subset L ⊂ {1, · · · , 2t + 1} with |L| = t + 1, take eRL ∈ Z[T ] is the resultant of (s−t) homogeneous polynomials eQj, j ∈ {2t+2, , s}∪L Since

Qj

j∈L are in t−subgeneral position with respect to X, there exists z0 ∈ D such that (s − t) homogeneous polynomials Q2t+2, , Qs, and Qj(z0), j ∈

L have non-trivial common solutions in Cn+1 (note that X is defined by the polynomials Qi ∈ C[x0, , xn], i ∈ {2t + 2, , s}) This means that e

RL( , akI, )(z0) 6= 0

By the assumption, for each j ∈ {1 , 2t + 1}, the set Aj := Q−1j (f ) does not depend on the mapping f ∈ F

Set

L⊂{1,··· ,2t+1},#L=t+1

{z ∈ D : eRL(· · · , akI, · · · )(z) = 0}

Then E := (∪2t+1i=1 Ai) ∪ A is a thin analytic subset of D

Let {fk}∞

k=1 ⊂ F be an arbitrary sequence

For any fixed point z0 ∈ D \ E, there exists an open ball B(z0, ) in D \ E such that

fk−1(Qi) ∩ B(z0, ) = ∅, for allk ≥ 1, andi ∈ {1, , 2t + 1} (3.1)

Since Qi( ˜fk) 6= 0 on B(z0, ), we get that B(z0, ) ∩ I(fk) = ∅ This implies, {fk|B(z0,)}∞

k=1⊂ Hol(B(z0, ), CPm)

We now prove that {fk|B(z0,)}∞

k=1 is a normal family on B(z0, ) In-deed, suppose that {fk|B(z0,)}∞

k=1 is not normal on B(z0, ), then by Lemma 3.3, there exist a subsequence (again denoted by {fk|B(z0,)}∞

k=1) and p0 ∈

B(z0, ), {pk}∞k=1 ∈ B(z0, ) with pk → p0, {ρk} ⊂ (0, +∞) with ρj → 0, Euclidean unit vectors {uk} ∈ Cm such that the sequence of holomorphic maps

gk(ξ) := fk(pk+ ρkukξ) : ∆rk → CPn, (rk → ∞) converges uniformly on compact subsets of C to a nonconstant holomor-phic map g : C → CPn Then, there exist reduced representations ˜gk = (gk0, · · · , gkn) of gk and a representation ˜g = (g0, · · · , gn) of g such that {˜gki} converges uniformly on compact subsets of C to ˜gi This implies that Qj(pk+

ρkukξ)(˜gk(ξ)) converges uniformly on compact subsets of C to Qj(p0)(˜g(ξ))

By (3.1) and Hurwitz’s theorem, for each j ∈ {1, , 2t + 1} we have

Img ∩ Qj(p0) = ∅, or Img ⊂ Qj(p0)

Here, we identify the polynomial Qj(p0) ∈ C[x0, · · · , xn] with the hypersur-face in CPn defined by Qj(p0)

It is clear that Q1(z0), , Q2t+1(z0) are in t−subgeneral position with respect

to X, sicne p0 6∈ A

Since, Imfk ⊂ X, for all k ≥ 1, we get that Img ⊂ X Without loss of generality, we may assume that Img ⊂ Qj(p0) for 1 ≤ j ≤ v and Img ∩ Q(z0) = ∅ for j ∈ {v + 1, , 2t + 1} (we take v = 0 for the case where Img ∩ Qj(z0) = ∅ for all j ∈ {1, , 2t + 1})

We have Img ⊂ M := X ∩ (∩v

j=1Qj(p0))

Case 1: v is even, v = 2`

We consider 2(t − `) + 1 hypersurfaces Qv+1(z0), , Q2t+1(z0) For any subset T ⊂ {v + 1, , 2t + 1}, with #T = (t − `) + 1, we have M ∩ (∩j∈TQj(z0)) = X ∩ (∩j∈TQj(z0)) ∩ (∩v

i=1Qi(z0)) = ∅ (note that #(T ∪ {1, , v}) = t − ` + 1 + v = t + 1 + ` ≥ t + 1) This means that 2(t − `) + 1 hypersurfaces Qv+1(z0), , Q2t+1(z0) are in (t−`)− subgeneral position with respect to M On the other hand, Img ⊂ M \ (∪2t+1j=v+1Qj(z0)) Hence, by Lemma 3.4, g is constant; this is a contradiction Therefore, {fk|Uz0}∞

k=1 is a normal family on B(z0, )

Case 2: v is old, v = 2` + 1

We consider 2(t − ` − 1) + 1 hypersurfaces Qv+1(z0), , Q2t(z0) For any subset T ⊂ {v + 1, , 2t}, with #T = t − `, we have M ∩ (∩j∈TQj(z0)) =

X ∩ (∩j∈TQj(z0)) ∩ (∩v

i=1Qi(z0)) = ∅ (note that #(T ∪ {1, , v}) = (t −

`) + v = t + 1 + ` ≥ t + 1) This means that 2(t − ` − 1) + 1 hypersurfaces

Qv+1(z0), , Q2t(z0) are in (t − `)−subgeneral position with respect to M

On the other hand, Img ⊂ M \ (∪2tj=v+1Qj(z0)) Hence, by Lemma 3.4, g is constant; this is a contradiction Hence, {fk|Uz0}∞

k=1 is a normal family on B(z0, )

By the usual diagonal argument, we can find a subsequence (again de-noted by {fk}∞

k=1) which converges uniformly on compact subsets of D \ E

to a holomorphic map f, Imf ⊂ X

By the assumption, ∩2t+1j=1 Sj is an analytic set of codimension ≥ 2 It

is clear that I(fk) ⊂ Sj for all k ≥ 1 and j ∈ {1, , 2t + 1} Therefore, I(fk) ⊂ ∩2t+1j=1Sj for all k ≥ 1 and j ∈ {1, , 2t + 1} This means that {fk}k≥1 are holomorphic on D \ (∩2t+1j=1 Sj)

For any fixed point z∗ ∈ D \ (∩2t+1

j=1 Sj), there exist an open ball B(z∗, ρ) ⊂

D \ (∩2t+1j=1 Sj) and an index j0 ∈ {1, , 2t + 1} such that Sj 0 ∩ B(z∗, ρ) = ∅ This means that

Q−1j0 (fk) ∩ B(z∗, ρ) = ∅ (3.2)

We define holomorphic mappings {Fk}∞

k=1 of B(z∗, ρ) into CPn+1 as follows: for any z ∈ B(z∗, ρ), if fk has a reduced representation ˜fk = (fk0, · · · , fkn)

on a neighborhood Uz ⊂ B(z∗, ρ) then Fk has a reduced representation ˜Fk = (fd

k0, · · · , fd

kn, Qj0( ˜fk)) on Uz Let Hi (i = 0, · · · , n) be hyperplanes in CPn

defined by

Hi = {(w0 : · · · wn)|wi = 0}

and let Hi (i = 0, · · · , n + 1) be hyperplanes in CPn+1 defined by

Hi = {(w0 : · · · wn+1)|wi = 0}

It is easy to see that {Fk} converges uniformly on compact subset of B(z∗, ρ)\

E to a holomorphic map F of B(z∗, ρ) \ E into CPn+1, and if f has a reduced representation ˜f = (f0, · · · , fn) on an open subset U ⊂ B(z∗, ρ) \ E then F has reduced representation ˜F = (fd

0, · · · , fd

n, Qj0( ˜f )) on U Since

f is holomorphic on B(z∗, ρ) \ E, there exists i0 (0 ≤ i0 ≤ n) such that

Hi 0( eF ) ≡ Hi 0( ef ) 6≡ 0 on B(z∗, ρ) \ E Then there exists k0 > 0 such that

Hi0( eFk) ≡ Hi0( efk) 6≡ 0 on B(z∗, ρ) \ E for all k > k0

Since Qj0( ef ) 6≡ 0 on B(z∗, ρ) \ E, we have Hn+1( eF ) 6≡ 0 on B(z∗, ρ) \ E On the other hand, by (3.2), for all k ≥ 1, we have

Fk−1(Hn+1) = fk−1(Qj0) ∩ B(z∗, ρ) = ∅

Therefore, by Lemma 3.2, {Fk} is meromorphically convergent on B(z∗, ρ) This implies that the sequence of divisors {ν(F

k ,Hi0)} converges on B(z∗, ρ), and hence {ν(fk,Hi0)} converges on B(z∗, ρ) By again Lemma 3.2, {fk} mero-morphically converges on B(z∗, ρ), for any z∗ ∈ D \ (∩2t+1

j=1 Sj) Hence, {fk} meromorphically converges on D \ (∩2t+1j=1 Sj) On the other hand, ∩2t+1j=1 Sj is

an analytic set of codimension ≥ 2 Hence, by Corollary 3.1, {fk} meromor-phically converges on D Then F is a meromormeromor-phically normal family on D

Proof of theorem 1.2 Let {fk}∞

k=1 ⊂ F be an arbitrary sequence Since {H∗

jf, f ∈ F } is a (holomorphically) normal family, without loss of generality,

we may assume that, for each j ∈ {1, , 2t + 1}, the sequence {Hjf∗

k}∞ k=1

converges uniformly on every compact subset of D to a holomorphic map L∗j with the reduced representation (bj0 : · · · : bjn) Let Lj (j = 1, , 2t + 1) be moving hyperplanes defined by homogeneous polynomials bj0x0+ · · · + bjnxn Since

D(H1fk, , H(2t+1)fk)(z) ≥ δ0 for all z ∈ D, k ≥ 1,

we have D(L1, , L2t+1)(z) ≥ δ0 for all z ∈ D This means that L1, , L2t+1 are in pointwise t−subgenral position in CPn

By the assumption, for each j ∈ {1 , 2t + 1}, the set Sj := {z : 1 ≤

ν(f,Hjf) ≤ mj} does not depend on the mapping f ∈ F

We now prove that:

Claim 1: There exists a subsequence of {fk}∞

k=1 (again denoted by {fk}∞

k=1) that converges uniformly on compact subset of D \(∪2t+1j=1 Sj) to a holomorphic map f on D \ (∪2t+1j=1 Sj) Here Sj is the colosure of Sj

For any fixed point z0 ∈ D \ (∪2t+1

j=1 Sj), we take an open ball B(z0, ) ⊂

D \ (∪2t+1j=1Sj) Then, for all j ∈ {1, , 2t + 1},

This implies, Hjfk(fk) has no zeros of multiplicity ≥ mj on B(z0, ) for all

k ≥ 1 and j ∈ {1, , 2t + 1} Since I(fk) ⊂ ∪2t+1j=1 Sj, we have that {fk}∞

k=1

are holomorphic on B(z0, )

We now prove that {fk|B(z0,)}∞

k=1 is a normal family on B(z0, ) In-deed, suppose that {fk|B(z0,)}∞

k=1 is not normal on B(z0, ), then by Lemma 3.3, there exist a subsequence (again denoted by {fk|B(z0,)}∞

k=1) and p0 ∈ B(z0, ), {pk}∞k=1 ∈ B(z0, ) with pk → p0, {ρk} ⊂ (0, +∞) with ρj → 0,

CPn Assume that {fk|D\S} meromorphically converges to a meromorphic mapping f of D \ S, then {fk} is meromorphically... {fi} be a sequence of meromorphic mappings of a domain D in Cm into CPn and let E be a thin analytic subset

of D Suppose that {fi} meromorphically...

j ∈ {1, , 2t + 1} and f ∈ F , where I(f ) is the set of all points of indeter-mination of f

Then F is a meromorphically normal family on D

Acknowledgements: This research is