LENGTH DISTORTION HOMEOMORPHISMS
VICTORIA STANCIU
Given a functionQ(z) of finite mean oscillation on a Riemann surfaceR,Q:R→ [1,∞], under suitable assumptions, we prove criteria for a family of finite length distortion homeomorphismsf : R →R0, between two homeomorphic Riemann surfacesRandR0 such that KI(z)≤Q(z), to be normal and closed.
AMS 2000 Subject Classification: 30C62.
Key words: FMO, FMD, FLD, Riemann surface, universal covering, normal fa- mily.
1. INTRODUCTION
LetD be a domain inC. A functionQ:D→Ris called of finite mean oscillation at a pointz0 ∈D, ifQ is integrable in a neighborhood ofz0 and
dQ(z0) = lim
ε→0
1
|B(z0, ε)|
Z Z
B(z0,ε)
|Q(z)−Qε(z0)|dxdy <∞, where
Qε(z0) = 1
|B(z0, ε)|
Z Z
B(z0,ε)
Q(z) dxdy
is the mean value of the functionQ(z) overB(z0, ε) ={z∈C:|z−z0|< ε}, see [4, p. 207], and|A|denotes the Lebesgue measure ofA⊂C. We calldQ(z0) the dispersionof the functionQatz0. We say that an L1loc(D) functionQ:D→R is offinite mean oscillationin the domainD, for shortQ∈FMO(D) or simply Q∈FMO, if Qhas finite dispersion at every pointz∈D.
Forz∈E ⊂C and a mappingϕ:E→C, we set
(1.1) L(z, ϕ) = lim sup
ζ→z, ζ∈E
|ϕ(ζ)−ϕ(z)|
|ζ−z| , and
(1.2) l(z, ϕ) = lim inf
ζ→z, ζ∈E
|ϕ(ζ)−ϕ(z)|
|ζ−z| .
REV. ROUMAINE MATH. PURES APPL.,54(2009),5–6, 575–583
We say that a mapping g : D → C (continuous) is said to be of finite metric distortion, abbr. g∈ FMD, ifg has the Lusin (N)-property and (1.3) 0< l(z, g)≤L(z, g)<∞a.e.
Recall that a mappingg:X→Y between measurable spaces (X,Σ, µ) and (X0,Σ0, µ0) is said to have the (N)-property ifµ0(g(S)) = 0 wheneverµ(S) = 0.
Similarly, g has the (N−1)-property ifµ(S) = 0 wheneverµ0(g(S)) = 0.
A path γ in C is a continuous mapping γ : ∆ → C, where ∆ is an interval in R.
Given a family of paths Γ inC, a Borel functionρ:C→[0,+∞] is called admissible for Γ, abbr. ρ∈adm Γ, if
Z
γ
ρ(z)|dz| ≥1 for each γ ∈Γ. Themodulus M(Γ) of Γ is defined as
M(Γ) = inf
ρ∈adm Γ
Z Z
ρ2(z) dxdy
interpreted as +∞ if adm Γ = ∅. Thus, every family Γ which contains a con- stant path is of infinite modulus. We say that a property P holds for almost every (a.e.) pathγ in a family Γ if the subfamily of all paths in Γ for which P fails has modulus zero.
If γ : ∆ → C is a locally rectifiable path, then there is the unique increasing length function lγ of ∆ onto a length interval ∆γ ⊂ R with a prescribed normalization lγ(t0) = 0 ∈∆γ,t0 ∈∆, such that lγ(t) is equal to the length of the subpath γ |[t0, t] of γ ift > t0, t ∈∆, and lγ(t) is equal to
−l(γ | [t, t0]) if t < t0, t ∈ ∆. Let g :γ(∆) → C be a continuous mapping, and suppose that the pathbγ =g◦γ also is locally rectifiable. Then there is a unique increasing function Lγ,g : ∆γ→∆
bγ such that (1.4) Lγ,g(lγ(t)) =l
bγ(t) for allt∈∆.
We say that a mappingg:D→Chas the (L)-property if (L1) for a.e. pathγ inD,γb=g◦γis locally rectifiable and the functionLγ,ghas the (N)-property;
and (L2) for a.e. path bγ ing(D), each lifting γ of bγ is locally rectifiable and the function Lγ,g has the (N−1)-property.
A pathγ inDis alifting of a pathbγ inCunderg:D→Cifbγ =g◦γ. Note that condition (L2) only applies to paths bγ which have a (maximal) lifting.
We say that a mapping g : D → C is of finite length distortion, abbr.
g∈ FLD, ifg is FMD and has the (L)-property.
LetDbe a domain inCand Q:D→[1,∞] a measurable function. Let FQ,δ(D) be the class of all FLD-homeomorphisms g : D → g(D) ⊂ C, suchb
that KI(z, g)≤Q(z) and
(1.5) h(Cb−g(D))≥δ >0 where h(A) is the spherical diameter ofA.
Given a mapping g : D ⊂ C → C with partial derivatives a.e., g0(z) denotes the Jacobian matrix ofgatz∈Dif it exists,J(z) =J(z, g) = detg0(z) the Jacobian of g at z, and |g0(z)| the operator norm of g0(z), i.e., |g0(z)|= max{|g0(z)w|:w∈C,|w|= 1}. Letl(g0(z)) = min{|g0(z)w|:w∈C,|w|= 1}.
The outer dilatation of g atz is defined as
KO(z) = KO(z, g) =
|g0(z)|2
|J(z, g)| ifJ(z, g)6= 0, 1 ifg0(z) = 0,
∞ otherwise,
the inner dilatation ofg atz as
KI(z) = KI(z, g) =
|J(z, g)|
l(g0(z))2 ifJ(z, g)6= 0, 1 ifg0(z) = 0,
∞ otherwise,
and themaximal dilatation of g atzas K(z) = K(z, g) = max(KO(z),KI(z)).
We remark that KI(z, g) = KO(z, g), (see [4, p. 6]).
We shall consider these classes of mappings on Riemann surfaces and give a definition of the FMO-functions on a Riemann surface R, by means of the universal covering (R,b Π, R), where Rb = B-the unit disk (but sometimes also a half-plane H = {zb ∈ C : Rez < a, ab ∈ R} or a vertical strip S = {bz ∈ C :s1 < Rez < sb 2, s1 and s2 ∈ R}) and Rb = C, according as R is of hyperbolic or parabolic type. Thus
FMO(R) ={Q:R→[1,∞]; Qb=Q◦Π∈FMO(R)}.b
LetRandR0be two homeomorphic Riemann surfaces, (R,b Π, R), (Rb0,Π0, R0)-their universal coverings, g:R→R0 a homeomorphism (embedding) and gb:Rb→Rb0 a lifting ofgto the universal coverings. LetQ∈FMO(R) andQb= Q◦Π∈FMO(R). By definitionb g is a FLD-homeomorphism(embedding) iff bg is a FLD-homeomorphism (embedding), z = Π(z). The condition Kb I(z, g)≤ Q(z) holds iff KI(bz,bg)≤Q(b z).b
In the following we shall work with mappings which have partial deriva- tives a.e. In order to reduce the proof of some statements about Riemann surfaces to the case of plane domains, we use [1, Proposition 1.2] which we shall call
Lemma 1.1. Let Φ be a family of homeomorphisms g:R→R0 between the Riemann surfaces R and R0, Φb a family consisting of at least a lifting gb: Rb → Rb0 with respect to the universal coverings (R,b Π, R) and (Rb0,Π0, R0) for every g∈Φ.
(i) If Φb is normal, then Φ is normal.
(ii)If Φb is normal and closed, then Φ is also normal and closed.
Theorem 1.1. Let D be a domain in C and {fm} a sequence of FLD- homeomorphisms of D into C with a.e. KI(z, fm) ≤Q(z) ∈L1loc, converging locally uniformly to a mappingf :D→C. Thenf is either anFLD-embedding or f ≡const. in D.
Proof. First step. As a locally uniform limit of continuous mappings,f is continuous. Let f 6= const. Let us first show thatf is a discrete mapping.
Indeed, iff is not discrete, then there is a pointz0∈Dand a sequencezk∈D, zk 6= z0, k = 1,2, . . . , such that zk → z0 as k → ∞ with f(zk) = f(z0).
Note that the set E0 = {z ∈ D : f(z) = f(z0)} is closed in D because f is continuous. Note also that E0 does not coincide with D because f 6= const.
Thus we can replace z0 by a boundary non-isolated point of the setE0. Without loss of generality we may assume thatz0 = 0,fm(0) =f(0) = 0, B ⊂D,B-the unit disk, and there is at least one pointζ0∈Bwheref(ζ0)6= 0.
By continuity of the spherical metric, h(fm(ζ0),0)≥ δ0/2, ∀m ≥ M0,where δ0=h(f(ζ0),0)>0.
SinceB is a compact set inD and fm→f uniformly in B, we have h(Cb−fm(B))≥δ1/2, ∀m≥M1,
where δ1 =h(Cb−f(B)).
Setting δ = min{δ0/2, δ1/2} and M = max{M0, M1}, by Theorem 4.4, p. 89 in [4] we have |fm(z)| ≥ ψ(|z|), ∀m ≥ M for all z ∈ B(0, r) and r = min|ζ0|
2 ,1− |ζ0| , whereψ is a strictly increasing function withψ(0) = 0 depending on kQk1, and δ only. Thus
(1.6) |f(z)| ≥ψ(|z|), ∀z∈B(0, r).
Then, in particular, 0 =|f(zk)| ≥ψ(|zk|),∀k≥k0, and, consequently,ψ(rk) = 0 for rk=|zk| 6= 0,k≥k0. This contradiction shows thatf is discrete.
Now, let us show that f is injective in D. Indeed, assume that there exist z1, z2 ∈ D, z1 6= z2, with f(z1) = f(z2). Let z2 ∈/ B(z1, t) ⊂ D for all t∈(0, t0]. Then everyfm(∂B(z1, t)), t∈(0, t0], separatesfm(z1) fromfm(z2) and, consequently, h(fm(z1), fm(∂B(z1, t)))< h(fm(z1), fm(z2)).Hence (1.7) h(f(z1), f(∂B(z1, t)))≤h(f(z1), f(z2)).
Since f(z1) = f(z2), by (1.7), for every t ∈ (0, t0] there is a point zt ∈
∂B(z1, t) such that f(zt) = f(z1). The latter contradicts the discreteness of
the mapping f. The continuity of the inverse mapping f−1 also follows from (1.6). Thus, f is a homeomorphism.
Second step. We first show that given a disk B, B ⊂ D, the partial derivatives of the mappings fm are uniformly bounded in L1(B). Indeed, in B we have
|∂fm(z)| ≤ |J(z, fm)|1/2· |fm0 (z)|
|J(z, fm)|1/2 = K1/2O (z, fm)· |J(z, fm)|1/2 and, by the Schwarz inequality,
k∂fmk1 ≤ kKO,mk1/21 · kJmk1/21 ≤ kQk1/21 · kJmk1/21 ,
where Jm(z) is the Jacobian of fm and k · k1 denotes the L1 norm in B. Now, kJmk1 ≤ |fm(B)| → |f(B)| by uniform convergence, thus the uniform boundedness of the partial derivatives of fm in B follows. Then f has finite partial derivatives a.e. in B and applying Theorem 3.1 (Gehring-Lehto) in [3, p. 128] we deduce that f is differentiable a.e. in B.
Let us remind Theorem 24.7 in [5, p. 85]. Namely, a homeomorphism f :D→ D0 ⊂C satisfies the (N)-property iff µ0(f F) = 0 for every compact set F ⊂Dsuch thatµ(F) = 0.
a) Here, fm is an FLD-homeomorphism, hence the (N)-property holds.
It follows that|fmF|= 0 for every compact setF ⊂Dsuch that|F|= 0. But fm →f locally uniformly, hence|fmF| → |f F|, thus|f F|= 0. It follows that f has (N)-property.
b) fm → f locally uniformly iff this convergence is continuous, i.e., fm(zm)→f(z0) ifzm→z0; see [2, p. 268]. Sincef is injective, we deduce that fm−1 →f−1 continuously, hence locally uniformly. Becausefm−1 →f−1 locally uniformly and fm−1 has the (N)-property, in turn f−1 has the (N)-property, hence f has the (N−1)-property. By Corollary 8.1 in [4, p. 150]f is FMD.
Lγ,f has the (N) and (N−1)-properties. Indeed, from fm → f locally uniformly, we deduce that l
bγm → l
γb locally uniformly. Therefore, since lγ is a surjection, we have Lγ,fm → Lγ,f locally uniformly. It follows as in a) that Lγ,f verifies (N) and (N−1), hence f has the property (L).
2. NORMALITY CRITERIA
The key idea of the proofs below consists in lifting the problem to the universal coverings and proving the results for the universal coverings.
Theorem 2.1. Let R and R0 be two homeomorphic Riemann surfaces, where R0 is non-conformally equivalent to either Cb or C, z0∈R, z00∈R0 and Q∈FMO(R). Then any family Φ of FLD-homeomorphisms g:R →R0 with
KI(z, g)≤Q(z) and
(2.1) g(z0) =z00
is normal.
Proof. If R0 is conformally equivalent to B, take R0 = B; then Φ con- sidered as a family of embeddings g:R →Cb is normal (since h(Cb−g(R)) = h(Cb−B) ≥ δ, thus Φ is included in FQ,δ(R) and one can use Theorem 8.22 in [4, p. 164]). It remains to prove the normality of Φ with respect to the distance d0 on R0. Let {gn} be a sequence in Φ and {gni} a sequence l.u.
convergent in R to g0 : R → B with respect to the spherical metric h. By Theorem 1.1, g0 is either a constant z00 ∈B or an embedding, in which case g0(R) is an open set, hence included inB. Now, one easily verifies that {gni} l.u. converges to g0 with respect tod0.
We shall use Lemmas 1.1 and 2.1 in what follows.
Lemma 2.1. Under the hypotheses of Theorem 2.1, let (R,b Π, R) and (Rb0,Π0, R0)be the universal coverings ofRandR0. Choose a pair of pointszb0∈ Π−1(z0)andbz00 ∈Π0−1(z00)and denote byΦb a family of FLD-homeomorphisms gb:Rb →Rb0 with KI(z,b bg) ≤Q(b bz), where bg is the lifting of a homeomorphism g∈Φ normalized by the condition bg(zb0) =zb00. Then Φb is normal.
Proof. Case1◦. R0 of hyperbolic type,Rb0=B.
Φ is normal as before in Theorem 2.1, the caseb R0=B.
Case2◦. R0 of parabolic type,Rb0=C.
Subcase 2◦.1. R and R0-tori, henceRb=C, too.
Suppose R = C/Zω1 +Zω2 and R0 = C/Zω10 +Zω20 with ωj, ωj0 ∈ C∗, j = 1,2, and Im
ω2
ω1
, Imω0 2
ω01
> 0. The covering groups G and G0 are generated by the translations Tj :zb7→ bz+ωj and Tj0 :bz0 7→ bz0+ω0j, j= 1,2, respectively. Without loss of generality we can take zb0 =bz00= 0. If bg∈Φ, byb
(2.2) bgTbg−1 =T0,
the isomorphism θbg : G → G0 associates with every T ∈ G, T : zb 7→ zb+ mω1 +nω2, the translation T0 = bgTgb−1 ∈ G0, T0 : bz0 7→ zb0+m0ω10 +n0ω02, m, n, m0, n0 ∈Z, thus (2.2) becomes
(2.3) bg(zb+mω1+nω2) =bg(z) +b m0ω10 +n0ω02, which can be also written in the form
(2.4) bg(bz+mω1+nω2) =bg(bz) +m0bg(ω1) +n0bg(ω2),
by emphazing the generatorsTj0 =θbg(Tj) ofG0,j = 1,2,Tj0:bz0 7→bz0+m0jω10+ n0jω20, m0j, n0j ∈ Z, det
m01 m02 n01 n02
= 1, since gb is sense-preserving. Remark thatbginduces byθbg a linear bijection between the two latticesZω1+Zω2 and Zω01+Zω02.
To prove that Φ is normal with respect tob h, note that the families Φ∗ of the restrictions g∗ = bg | C∗, and Φ∗∗ of the restrictions g∗∗ = bg | [C−(Zω1+Zω2)∗], viewed as FLD-homeomorphisms intoCb, are normal. In- deed,h(Cb−g∗(C∗))≥δ, h(Cb−g∗∗(C−(Zω1+Zω2)∗))≥δ and hence the fam- ilies Φ∗and Φ∗∗are included, respectively, inFQ,δ(C∗), FQ,δ(C−(Zω1+Zω2)∗) and one can use [4, Theorem 8.22, p. 164]). Therefore, Φ is normal inb C = C∗∪[C−(Zω1+Zω2)∗] with respect toh.
Subcase 2◦.2. R0 = C∗, however R = C∗, B(0, r)∗ with 1 < r < ∞, or A = {z : r1 < |z| < r2} with 0 < r1 < 1 < r2 < ∞, hence Rb is C, H ={zb: Rez <b logr} orS={bz: logr1 <Rez <b logr2}, respectively.
Consider the universal coverings (R,b exp, R) and (C,exp,C∗) and choose zb0 =zb00= 0. By (2.2), for any bg∈Φ we haveb
(2.5) bg(bz+mπi) =bg(bz)±2mπi, m∈Z.
The familyΦ is normal with respect tob h,h(Cb−bg(C∗))≥δ, h(Cb−bg(C− {±2πi}))≥δ, since at three fixed points, e.g., 0,±2πi, every bg∈Φ takes theb values 0, ±ε2πi (ε=±1).
In both subcases 2◦.1 and 2◦.2, normality of Φ with respect tob d0 follows by arguments similar to those in Theorem 2.1, case R0 = B. Normalization implies that bg0 is not the constant∞. Asgb0 is an FLD-embedding by Theo- rem 1.1, we deduce from (2.4) and (2.5), respectively, that ∞∈/ bg0(R). Henceb gb0(R)b ⊂Rb0. Finally, one verifies that bgn u.converges to gb0 on every compact M ⊂Cwith respect to both distancesh and d0, simultaneously.
The proofs of the following results are similar to those in [6].
Theorem 2.2. Let R and R0 be two homeomorphic Riemann surfaces, R0 not conformally equivalent to either CorCb,z0 an arbitrary point in Rand E0 a compact subset of R0. Let Q be a function in FMO(R). Denote by Φ a family of FLD-homeomorphisms g:R→R0 withKI(z, g)≤Q(z) and
(2.6) g(z0)∈E0.
If Φ6=∅ thenΦ is normal.
Theorem 2.2 follows from Lemma 2.2 below and Lemma 1.1.
Lemma 2.2. Under the hypotheses of Theorem 2.2, let (R,b Π, R) and (Rb0,Π0, R0) be the universal coverings of R and R0, respectively, let bz0 ∈ Π−1(z0) and Eb0 be a compact set in Rb0 with Π0(Eb0) ⊃ E0. The family Φb of FLD-homeomorphisms bg : Rb → Rb0, Qb = Q◦Π ∈ FMO(R), whereb bg is the lifting of g∈Φ normalized by the condition
(2.7) bg(bz0)∈Eb0, is normal.
Remark 2.1. Theorems 2.1 and 2.2 does not hold in the casesR0 =C or Cb sinceh(Cb−R0) is not strictly greater than 0.
In the following we replace condition (2.6) by the conditions
(2.60) g(E)⊂E0
or
(2.600) g|E :E →E0 is a homeomorphism,
where z0 ∈ R is replaced by a compact set E ⊂ R, E 6= ∅. The following result holds.
Theorem2.20. Let R, R0, E0 be as in Theorem 2.2, and let E be a com- pact subset of R. If a family Φ0 (Φ00) of FLD-homeomorphisms g : R → R0, Q ∈ FMO(R) which verify (2.60) ((2.600), respectively) is nonempty, then it is normal.
3. COMPACTNESS We have
Theorem 3.1.Under the hypotheses of Theorem 2.1, if R0 is not confor- mally equivalent toCor B(R0can beCb), then the limit f of any l.u. convergent sequence {fm} ⊂Φ is an FLD-embedding.
To prove this theorem we establish
Lemma 3.1.Under the hypotheses of Lemma 2.1, if{fbm} ⊂Φb l.u. con- verges to f, thenb fbis an FLD-embedding.
Proof. By Theorem 1.1,fbis the constantzb00 or an FLD-embedding. Take T ∈ G different from the identity in G. Then it follows from fbmT =Tm0 fbm, that fbmTbz0 =Tm0 zb00, whereTm0 =θ
fbm(T). If fbis constant, sincefbmTbz0 tends to f Tb bz0, for m large Tm0 reduces to the identity and so does T, which is a contradiction.
Proof of Theorem 3.1. Let fbm : Rb → Rb0 be a lifting of fm such that fbm(bz0) = bz00. Since the family Φ is normal (Lemma 2.1), we can extract ab subsequence{fbmk}l.u. convergent inRbto an FLD-embeddingfb(Lemma 3.1).
Then Lemma 1.1 shows that f is an FLD-embedding.
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Received 25 January 2009 “Politechnica” University of Bucharest Faculty of Applied Sciences
313 Splaiul Independent¸ei 060032 Bucharest, Romania
victoriastanciu@yahoo.com
