# 1-Quasiconformal Mappings and CR Mappings on Goursat Groups

Authors: Qing Yan Wu; Zun Wei Fu
Journal: The Scientific World Journal
 (2014)

---

## Abstract

We show that 1-quasiconformal mappings on Goursat groups are CR or anti-CR mappings. This can reduce the determination of 1-quasiconformal mappings to the determination of CR automorphisms of CR manifolds, which is a fundamental problem in the theory of several complex variables.


---

## Body

## 1. Introduction

The well-known Liouville theorem tells us that 1-quasiconformal mappings onℝn are elements of group SO(n+1,1). Quasiconformal mappings on the Heisenberg group were introduced by Mostow [1] in his study of rigidity theorem. Since then, quasiconformal mappings on nilpotent Lie group and on more general Carnot-Carathéodory spaces had drawn more and more concern. Korányi and Reimann [2] and Korányi and Reimann [3] showed that C4 orientation preserving 1-quasiconformal mappings between domains of Heisenberg group ℍn are CR or anti-CR mappings and are obtained by the restriction of SU(n+1,1) group actions (see also [4]). Cowling et al. [5] extended Liouville theorem to conformal mappings on G/P, where G is a semisimple Lie group and P is a minimal parabolic subgroup of G. Ottazzi and Warhurst [6, 7] obtained a Liouville type theorem for all Carnot groups which sated that 1-quasiconformal maps form finite dimensional Lie groups. Xie [8] investigated 1-quasiconformal mappings on Carnot groups with reducible first layer equipped with left-invariant sub-Riemannian Carnot metrics and showed that each 1-quasiconformal map defined on a domain of such groups is the restriction of the composition of a standard dilation, a left translation, and an isometric graded isomorphism. In [9], we showed the CR property of 1-quasiconformal mappings on the Engel group and obtained the identity component of the group of 1-quasiconformal mappings. For other deformations of CR mappings between the nilpotent Lie group of step two and their Beltrami equations, one can refer to [10]. In this paper, we will investigate the 1-quasiconformal mappings on Goursat groups, which are stratified nilpotent Lie groups including Engel groups and a large class of high step Carnot groups.AGoursat group is a nilpotent group endowed with a Goursat distribution D. A distribution D of corank s≥2 on a manifold M isGoursat if the subsheaves Di of the tangent bundle defined inductively by Di+1=[Di,Di] (i=1,2,…,s;D1=D;[Di,Di] denotes the sheaf of vector fields generated by Di and the Lie brackets [X,Y],X,Y∈Di) correspond to distributions; that is, they have constant rank, and this rank is rank⁡Di+1=rank⁡Di+1, i=1,2,…,s [11]. By a distribution of rankd we understand any subbundle of linear dimension d in the tangent bundle TM; its corank is dim⁡M-d.Our model of Goursat groups isGn in [12], which takes ℝn+1 as the underlying space with the group multiplication
(1)(x0,x1,…,xn)(x0′,x1′,…,xn′)=(12x0+x0′,x1+x1′,x2+x2′+x0x1′,…,xn+xn′+x0xn-1′+12x02xn-2′+⋯+1(n-1)!x0n-1x1′).
It is easy to see that its Lie algebra of left-invariant vector fields is given by
(2)X0=∂∂x0,Xj=∂∂xj+x0∂∂xj+1+⋯+x0n-j(n-j)!∂∂xn,j=1,…,n.
Then X0,X1,…,Xn span the tangent space everywhere and the only nontrivial commutation relations among these vector fields are
(3)[X0,Xj]=Xj+1,j=1,…,n-1.
We use 𝔤 to denote the Lie algebra and it splits into the direct sum
(4)𝔤=V1⊕V2⊕⋯⊕Vn,
where the vector space V1 is spanned by the fields X0,X1 and the space Vj,j=2,…,n, is spanned by Xj, respectively. It is clear to see that when n=3, Gn is Engel group.Letdc(·,·) be the Carnot-Carathéodory distance on the Goursat group. A homeomorphism f:Ω→Ω′ between domains on the Goursat group G isquasiconformal if
(5)H(p)=limsup⁡r→0H(p,r)=limsup⁡r→0max⁡dc(p,q)=r⁡dc(f(p),f(q))min⁡dc(p,q)=r⁡dc(f(p),f(q)),p∈Ω,
is uniformly bounded. If in addition ∥H∥∞=sup⁡p∈Ω⁡H(p)≤K the homeomorphism f is K-quasiconformal.Onℝ2, a 1-quasiconformal mapping is holomorphic or antiholomorphic. If we identify the Heisenberg group with a quadratic hypersurface in ℂn, 1-quasiconformal mapping on the quadratic hypersurface is the restriction of a holomorphic or antiholomorphic mapping on ℂn [3]. This can be viewed as the higher dimensional case of the holomorphicity of 1-quasiconformal mapping. This phenomenon happens for Goursat group.ACR manifold is a smooth manifold M equipped with a distribution H⊂TM of even rank 2n and a complex structure as endomorphisms Jp:Hp→Hp with Jp2=-id. Denote
(6)T1,0M=ker⁡(Jp--1id),T0,1M=ker⁡(Jp+-1id).
A CR manifold is supposed to satisfy the following integrability condition: [Z1,Z2]∈C∞(T1,0M), if Z1,Z2∈C∞(T1,0M). The number n is calledCR dimension ofM and k=dim⁡M-2n is called thecodimension ofM. A smooth mapping f:M1→M2 is calledCR if
(7)f*T1,0M1⊂T1,0M2.
CR manifolds appear naturally as embedded real submanifolds of complex manifolds. The distribution H is then defined as H=TM∩iTM and J is the restriction of the complex structure in the ambient complex manifold to H. We recall that, for a manifold M={ρ1=0,ρ2=0,…,ρk=0} with ∂¯ρ1∧∂¯ρ2∧⋯∧∂¯ρk≠0, the induced CR structure on M is given by T1,0M=T1,0ℂn∩ ℂTM= {Z∈T1,0ℂn∣Zρ1=0,Zρ2=0,…,Zρk=0}, and dim⁡T1,0M=n-k.The Goursat groupGn can be realized as the real submanifold of ℂn:
(8)Sn={(z,w2,…,wn)∈ℂn∣Im⁡wk=-12kk!(zk+z¯k)∑j=1k-1‍-12k∑j=1k-1‍1j!(k-j)!zjz¯k-j,∑j=1k-1‍k=2,…,n},
where z=x+iy, wj=uj+ivj, j=2,…,n, by the diffeomorphism T:Sn→Gn defined by
(9)T(z,w2,…,wn)=(z,Rew2,…,Rewn).
It is easy to check that
(10)Z=∂∂z-iRez∂∂w2-⋯-i(n-1)!(Rez)n-1∂∂wn,
where ∂/∂z=(1/2)((∂/∂x)-i(∂/∂y)), ∂/∂wj=(1/2)((∂/∂uj)-i(∂/∂vj)), j=2,…,n, is the complex tangent vector of Sn, and we can define the standard CR structure on Sn by T1,0Sn=ℂZ and dim⁡T1,0Sn=1. We can also define quasiconformal mappings on Sn by (5) with respect to its Carnot-Carathéodory distance defined in Section 2. There is one to one corresponding between 1-quasiconformal mappings on Gn and Sn via the diffeomorphism T. Therefore, if we want to study 1-quasiconformal mappings on Gn, we can just study the 1-quasiconformal mappings on Sn. For further details see Section 2.Denote(11)𝒫={(x0,x1,…,xn)∈Gn∣x2=x3=⋯=xn=0}.
We show that 1-quasiconformal mappings between domains on Goursat group Gn whose Pansu differentials preserve 𝒫 are CR or anti-CR.Theorem 1.
LetΩ be a domain in Sn. Suppose that f=(f1,…,fn):Ω→f(Ω) is a 1-quasiconformal mapping with Df~(p) preserving 𝒫; then f is CR or anti-CR; that is, either
(12)Z¯fk=0,k=1,2,…,n,
or
(13)Zfk=0,k=1,2,…,n,
where f~=T∘f∘T-1 and p is some P-differential point.Remark 2.
By this theorem the determination of 1-quasiconformal mappings is reduced to the determination of CR automorphisms of a submanifold inℂn. According to the extension theorem (Theorem 1.4 in [13]), locally, the smooth CR automorphisms of Sn are holomorphic automorphisms of Sn, which is also an interesting problem.Remark 3.
In the cases of Heisenberg group and Engel group, the condition ofDf~(p) preserving 𝒫 in Theorem 1 can be omitted. However, by a similar method as in those cases, we cannot remove it, since it is inconvenient to use the expressions of Baker-Campbell-Hausdorff formula for groups of higher step.


## 2. Preliminaries

There is a natural homogeneous norm∥·∥ on Gn, given by
(14)∥(x0,x1,…,xn)∥2n!=(x02+x12)n!+∑k=2n‍|xk|2n!/k,
and an associated pseudometric d, given by
(15)d(p,q)=∥q-1p∥,
for p,q∈Gn. The metric is left-invariant and is related to the dilations by the formula
(16)d(δcp,δcq)=cd(p,q),
where
(17)δc(x0,x1,…,xn)=(cx0,cx1,c2x2,…,cnxn),c∈ℝ+,
denotes the dilation on Gn.Thehorizontal tangent space atp∈Gn is a subspace HpGn of the tangent space TpGn, and it is spanned by the vector fields X0(p) and X1(p). An absolutely continuous curve γ:[0,1]→Gn ishorizontal if its tangent vectors γ˙(t), t∈[0,1], lie in the horizontal tangent space Hγ(t)Gn. By Chow [14], any given two points p,q∈Gn can be connected by a horizontal curve.OnHpGn, we fix a quadric form 〈·,·〉G with respect to which the vector fields X0(p), X1(p) are orthonormal at every point p∈Gn and satisfy
(18)|αX0(p)+βX1(p)|G2=〈αX0(p)+βX1(p),αX0(p)+βX1(p)〉G=α2+β2,
where α,β∈ℝ. 〈·,·〉G is calledinfinitesimal Carnot-Carathéodory (C-C) metric. Then thelengthl(γ) of the curve γ is defined by
(19)l(γ)=∫01‍〈γ˙(t),γ˙(t)〉G1/2dt.TheCarnot-Carathéodory distance (C-C distance)dc(p,q) is the infimum of the lengths of all absolutely continuous horizontal curves connecting p and q:
(20)dc(p,q)=inf⁡γl(γ).
According to [15], this metric is equivalent to the pseudometric; that is,
(21)C1d(x,y)≤dc(x,y)≤C2d(x,y),∀x,y∈Gn,
for suitable constants C1,C2.For the manifoldSn, the horizontal tangent space at x∈Sn is HxSn=span⁡ℝ{ReZ(x),Im⁡Z(x)}. On HxSn, we fix a quadric form 〈·,·〉S with respect to which 2ReZ(x), 2Im⁡Z(x) are orthonormal at every point x∈Sn and satisfy
(22)|aReZ(x)+bIm⁡Z(x)|S2=〈aReZ(x)+bIm⁡Z(x),aReZ(x)+bIm⁡Z(x)〉S=a2+b24,
where a,b∈ℝ. The C-C distance on Sn can be defined as above.By definition (8), Sn in ℂ×ℂn-1 corresponding to Gn is defined by the equation
(23)ρ1=ρ2=⋯=ρn-1=0,
where
(24)ρ1(z,w2,…,wn)=w2-w¯22i+14Rez2+14|z|2,ρ2(z,w2,…,wn)=w3-w¯32i+18Rez2z¯+124Rez3,⋮ρn-1(z,w2,…,wn)=wn-w¯n2i+12n∑j=1n-1‍1j!(n-j)!zjz¯n-j+12nn!(zn+z¯n).
It is direct to check that
(25)∂¯ρ1∧∂¯ρ2∧⋯∧∂¯ρn-1≠0.
And the vector field Z defined by (10) satisfies
(26)Zρj=0,j=1,…,n-1.
That is, Z is a complex tangent vector field of Sn, and Sn is a CR manifold with the induced CR structure T1,0Sn=ℂZ. We obtain the following relation between Sn and Gn.Proposition 4.
S
n is CR diffeomorphic and isometric to Gn via T defined by (9); that is, T and T-1 are CR and isometric.Proof.
We can denote the point inGn as (z,x2,x3,…,xn) where z=x0+ix1, and then the group multiplication of Gn can be written in the following form:
(27)(z,x2,…,xn)(z′,x2′,…,xn′)=(z+z′,x2+x2′+(Rez)Im⁡z′,…,xn1(n-1)!+xn′+(Rez)xn-1′+12(Rez)2xn-2′+⋯+1(n-1)!(Rez)n-1Im⁡z′).
Then the complex horizontal space is spanned by ZG,Z¯G, where
(28)ZG=12(X0-iX1)=∂∂z-i2(Rez)∂∂x2-⋯-i2(n-1)!(Rez)n-1∂∂xn;
here ∂/∂z=(1/2)((∂/∂x0)-i(∂/∂x1)).
We claim that, under this diffeomorphismT defined by (9), the complex tangent vector fields Z,Z¯ of Sn are mapped onto ZG and Z¯G, respectively. In fact, for any f∈Ck(Gn) and q=(z,w2,…,wn)∈Sn,
(29)(T*Z)f(q)=Z(f∘T)(q)=[∂∂z-iRez∂∂w2-⋯-i(n-1)!(Rez)n-1∂∂wn][f(z,z¯,w2+w¯22,…,wn+w¯n2)]=∂f∂z-iRez2∂f∂x2-⋯-i2(n-1)!(Rez)n-1∂f∂xn=ZGf(T(q)).
Therefore,
(30)T*Z=ZG.
Similarly
(31)T*Z¯=Z¯G.
It is clear to see thatT-1:Gn→Sn is
(32)T-1(z,x2,…,xn)=(z,x2+iv2,…,xn+ivn),
where
(33)vk=-12kk!(zk+z¯k)-12k∑j=1k-1‍1j!(k-j)!zjz¯k-j,hhhhhhhhhhhhhhhhhhhhhlhhhhk=2,…,n.
Similarly, we find(34)T*-1ZG=Z,T*-1Z¯G=Z¯.
Hence, T,T-1 are CR and by (30), (31), and (34) together with (18) and (22), we get |T*X|G=|X|S for any X∈HqSn and |T*-1Y|S=|Y|G for any Y∈HpGn, so T,T-1 preserve C-C distance. This proves the proposition.


## 3.P-Differential of 1-Quasiconformal Mappings on Gn

The notion of differential of a map between domains in Carnot group is due to Pansu [16]. Let τp(h)=ph denote left-translation on Gn. The mapping f:Ω→Ω′ (Ω,Ω′ are domains in Gn) is calledp-differentiable atp∈Ω if, for c→0, the mappings
(35)δc-1∘τf(p)-1∘f∘τp∘δc
converge locally uniformly to a homomorphism Df(p) of the group Gn, which is calledPansu differential (P-differential).The Pansu differentialDf(p) is a Carnot homomorphism. Denote by Df*(p) the induced Lie algebra homomorphism of the group homomorphism Df(p). According to [17], Df*(p) is grading-preserving. As a result of [16] or [18], quasiconformal mappings between domains in Carnot group are P-differentiable a.e. The result is valid on the group Gn as it is a Carnot group.Define the bilinear formB on 𝔤 by
(36)[Xj,Xk]=B(Xj,Xk)X2,j,k=0,1.
Then by [X0,X1]=X2, the corresponding matrix is
(37)B=(01-10).Letf be a 1-quasiconformal mapping between domains in Gn; since Df*(p) is grading-preserving, then Df*(p) maps Vj  (j=1,…,n) onto itself, respectively. In particular V1=span⁡{X0,X1}, so for any X∈V1,
(38)Df*(p)X=SX,
for some 2×2 matrix S. Since V2=ℝX2, then
(39)Df*(p)X2=λX2
for some λ.Proposition 5.
The matrixS above can be written as the following form:
(40)S=(ccos⁡θcsinθ-λcsinθλccos⁡θ),
for some θ∈[0,2π), where c=|Df*(p)X|G for any X∈V1 with |X|G=1, λ2/c4=1; λ,θ,c depend on p∈Gn.Proof.
SinceDf*(p)X=SX,X∈V1, now suppose
(41)S=(s11s12s21s22);
then
(42)Df*(p)Xk=s1kX0+s2kX1,k=0,1.
BecauseDf*(p) is Lie algebra homomorphism, that is,
(43)[Df*(p)Xk,Df*(p)Xl]=Df*(p)[Xk,Xl],
together with Df*(p)X2=λX2, we find that
(44)B(s1kX0+s2kX1,s1lX0+s2lX1)=λB(Xk,Xl),
namely,
(45)STBS=λB.
By Lemma 3.3 in [19], for the 1-quasiconformal mapping f, we have
(46)max⁡X∈V1,|X|G=1⁡|Df*(p)X|Gmin⁡X∈V1,|X|G=1⁡|Df*(p)X|G=1,
where |·|G is defined by (18), that is,
(47)max⁡X∈V1,|X|G=1|SX|Gmin⁡X∈V1,|X|G=1|SX|G=1.
Therefore, there exists c>0, such that
(48)|SX|G=c,
for any X∈V1, |X|G=1. Then (1/c)S is an orthogonal matrix. Together with (45), we have
(49)BS=λc2SB,
that is,
(50)(01-10)(s11s12s21s22)=λc2(s11s12s21s22)(01-10).
Thus
(51)s21=-λc2s12,s22=λc2s11,s11=λc2s22,s12=-λc2s21.
Therefore, λ2/c4=1. Note that ((1/c)S)T((1/c)S)=I, where I is the 2×2 unit matrix; hence
(52)s112+s122=c2,s212+s222=c2,s11s21+s12s22=0.
From (51), (52) and setting s11=ccos⁡θ, θ∈[0,2π), we have (40). The result follows.We get the following relation betweenP-differential and the usual derivatives. For the corresponding result on Heisenberg group see p.31 of [3]. In the following, the letter C will be used to denote various constants, and the various uses of the letter do not, however, denote the same constant.Proposition 6.
Supposef is a quasiconformal mapping between domains in Gn with f(0)=0, and if f is P-differentiable at 0 and Df(0) preserves the plan 𝒫 in (11), then f is differentiable in the usual sense at 0 along the horizontal directions and
(53)Df*(0)∂xj=f*0∂xj,j=0,1,
where f*0∂xj=∑k=0n‍(∂fk/∂xj)(0)∂xk and Df*(0) is the P-derivative of f at 0.Proof.
Sincef is P-differentiable at 0, by definition, there exists 0<δ<1 such that, for c<δ and ∥(z,x2,…,xn)∥≤1,
(54)∥(Df(0)(z,x2,…,xn))-1×(δc-1∘τf(0)-1∘f∘τ0∘δc(z,x2,…,xn))∥=∥(Df(0)(z,x2,…,xn))-1×(δc-1∘f∘δc(z,x2,…,xn))∥<ɛ.
Recall that the homomorphism Df(0) commutes with δc; then for points (z/|z|,0,…,0) in particular, we have
(55)∥δc-1((Df(0)(cz|z|,0,…,0))-1f(cz|z|,0,…,0))∥=1c∥(Df(0)(cz|z|,0,…,0))-1f(cz|z|,0,…,0)∥<ɛ.
Therefore
(56)1|z|∥(Df(0)(z,0,…,0))-1f(z,0,…,0)∥<ɛ,hhhhhhhhhhhhhhhhhhhhhlhhhhhif|z|<δ.
Now compare this with Euclidean distances. SinceDf(0) preserves the plane𝒫, that is,
(57)Df(0)(z,0,…,0)=(z~,0,…,0)=(x~0,x~1,0,…,0)
for some z~=(x~0,x~1)∈ℝ2, then
(58)(Df(0)(z,0,…,0))-1=(x~0,x~1,0,…,0)-1=(-x~0,-x~1,x~0x~1,-x~022x~1,…,(-1)nx~0n-1(n-1)!x~1).
Assume that (x0′,x1′,x2′,…,xn′)=f(z,0,…,0); we get
(59)∥(Df(0)(z,0,…,0))-1f(z,0,…,0)∥2n!=∥(-x~0,-x~1,x~0x~1,-x~022x~1,…,(-1)nx~0n-1(n-1)!x~1)×(x0′,x1′,x2′,…,xn′)(x~022(-1)nx~0n-1(n-1)!)∥2n!=(|x0′-x~0|2+|x1′-x~1|2)n!+|x2′-x~0(x1′-x~1)|2n!/2+|x3′-x~0x2′+(-x~0)22(x1′-x~1)|2n!/3+⋯+|xn′-x~0xn-1′+(-x~0)22xn-2′+⋯+(-x~0)n-1(n-1)!(x1′-x~1)|2n!/n<ε2n!|z|2n!.
It follows that
(60)|x0′-x~0|2+|x1′-x~1|2<ε2|z|2,|x2′-x~0(x1′-x~1)|<ε2|z|2,|x3′-x~0x2′+(-x~0)22(x1′-x~1)|<ε3|z|3,⋮|xn′-x~0xn-1′+(-x~0)22xn-2′+⋯+(-x~0)n-1(n-1)!(x1′-x~1)|<εn|z|n.
Therefore,
(61)|x0′-x~0|<ɛ|z|,|x1′-x~1|<ɛ|z|,(62)|x2′|<ɛ2|z|2+ɛ|x~0||z|,|x3′|<ɛ3|z|3+|x~0||x2′|+(-x~0)22ɛ|z|⋮|xn′|<ɛn|z|n+|x~0||xn-1′|+(-x~0)22|xn-2′|+⋯+ɛ|(-x~0)n-1(n-1)!||z|.
From Lemma 3.3 in [19], Df(0) is Lipschitz; then by the equivalence of dc and d, we have
(63)∥Df(0)(z,0,…,0)∥≤CLipf(0)∥(z,0,…,0)∥,
for some C>0, namely,
(64)|x~0|≤∥Df(0)(z,0,…,0)∥≤CLipf(0)|z|=M1|z|.
Sincef is continuous, then when |z|<δ<1, we have
(65)|(x0′,x1′,x2′,…,xn′)|≤|f(z,0,…,0)|<M2,
where M2 is a positive constant. Now substituting (64) and (65) into (62) and then by iteration, we obtain
(66)|x2′|<ɛ2|z|2+ɛM1|z|2<Cɛ|z|2,|x3′|<ɛ3|z|3+|x~0||x2′|+(-x~0)22ɛ|z|<ɛ3|z|3+Cɛ|x~0||z|2+(-x~0)22ɛ|z|<Cɛ|z|3,⋮|xn′|<ɛn|z|n+|x~0||xn-1′|+(-x~0)22|xn-2′|+⋯+ɛ|(-x~0)n-1(n-1)!||z|<ɛ2|z|n+Cɛ|x~0||z|n-1+Cɛ(-x~0)22|z|n-2+⋯+Cɛ|(-x~0)n-1(n-1)!||z|<Cɛ|z|n.
Then by (61), for any |z|<δ<1, we have
(67)1|z||Df(0)(z,0,…,0)-f(z,0,…,0)|=1|z||(x~0,x~1,0,…,0)-(x0′,x1′,x2′,…,xn′)|=1|z|(|x0′-x~0|2+|x1′-x~1|2+∑k=2n‍|xk′|2)1/2<C1|z|(ɛ2|z|2+∑k=2n‍ɛ2|z|2k)1/2<Cɛ.
If we letz=(x0,0), then
(68)1|x0||Df(0)(x0,0,…,0)-f(x0,0,…,0)|=|Df(0)(1,0,…,0)-f(x0,0,…,0)x0|<Cɛ,
namely,
(69)Df(0)(1,0,…,0)=lim⁡x0→0f(x0,0,…,0)-f(0)x0=∂f∂x0(0).
Therefore,
(70)Df*(0)∂x0=∑j=0n‍∂fj∂x0(0)∂xj=f*0∂x0.
Similarly
(71)Df*(0)∂x1=∑j=0n‍∂fj∂x1(0)∂xj=f*0∂x1.
The proposition is proved.We need the following relation betweenP-differential and the usual tangential mapping. See Proposition 3.2 in [10] for the corresponding result on the Heisenberg group.Proposition 7.
Letf:Ω1→Ω2 be a quasiconformal mapping between domains in the Goursat group Gn. Suppose f is P-differentiable at p and Df(p) preserves the plan 𝒫 in (11); then
(72)f*(p)X(p)=(τf(p))*0[Df*(p)X(0)],
for X∈V1, where f*(p)X(p)=∑k=0n‍Xfk(p)∂xk and τq* are the tangential mappings of τq for q∈Gn.Proof.
Supposef is P-differentiable at p∈Gn; let F=τf(p)-1∘f∘τp; then F is also a quasiconformal mapping and F(0)=0. Moreover, by Chain rule (Lemma 3.7 in [19]), F is P-differentiable at the origin 0, and
(73)DF(0)=Dτf(p)-1(f(p))Df(p)Dτp(0).
Since Dτp is the identity automorphism, then
(74)DF(0)=Df(p).
Therefore, DF(0) preserves the plan 𝒫.
By Proposition6, we have
(75)DF*(0)∂xj=F*0∂xj,j=0,1.
From (2) we can see that X0(0)=∂x0 and X1(0)=∂x1. Therefore,
(76)DF*(0)Xj(0)=F*0Xj(0),j=0,1.
Consequently,
(77)f*(p)Xj(p)=(τf(p)∘F∘τp-1)*pXj(p)=(τf(p))*0F*0Xj(0)=(τf(p))*0[DF*(0)Xj(0)]=(τf(p))*0[Df*(p)Xj(0)],00000000000000000000j=0,1.
This completes the proof.We need the following proposition due to Capogna ([19], Theorem 1.1).Proposition 8.
Suppose thatΩ1 and Ω2 are open subsets of Carnot groups G1 and G2, and f:Ω1→Ω2 is 1-quasiconformal. Then f is smooth.Proof of Theorem1.
Letf be a 1-quasiconformal mapping satisfying the hypothesis in Theorem 1. By Proposition 4, T,T-1 defined by (9) and (32) are CR and isometric; then f~=T∘f∘T-1 is a 1-quasiconformal mapping between domains in Gn. Suppose p∈Gn is a P-differentiable point of f~; then Df~*(p) is grading-preserving; that is, Df~*(p)X2=λX2 for some λ, and Df~*X=SX,X∈V1. From Proposition 5, at p∈Gn, S has form (40), and λ=±c2, where c=|Df~*(p)X|G for any X∈V1 with |X|G=1.
In the complexified subspaceℂV1, the corresponding complex basis is ZG, Z¯G, where ZG is as (28). On the matrix level
(78)S⟶Sc=ΘSΘ-1,
with Θ=(1i1-i). Then
(79)Sc=(c2+λ2ce-iθc2-λ2ceiθc2-λ2ce-iθc2+λ2ceiθ),θ∈[0,2π).
We claim thatDf~*(p)ZG=ScZG, Df~*(p)Z¯G=ScZ¯G. In fact, by (28) and (40), we have
(80)Df~*(p)ZG=Df~*(p)[12(X0-iX1)]=12[cX0cos⁡θ-λcX1sinθ-i(cX0sinθ+λcX1cos⁡θ)]=12[ce-iθX0-iλce-iθX1]=12e-iθ[c(ZG+Z¯G)+λc(ZG-Z¯G)]=12e-iθ[(c+λc)ZG+(c-λc)Z¯G]=ScZG.
Similarly, Df~*(p)Z¯G=ScZ¯G.
From Proposition8, f~ is smooth, and therefore f=T-1∘f~∘T is also smooth. We write the 1-quasiconformal mapping f in the form
(81)f(z,w2,…,wn)=(f1,f2,…,fn)(z,w2,…,wn)=(ω1,ω2,…,ωn)
with
(82)Im⁡fk=-12kk!(f1k+f¯1k)-12k∑j=1k-1‍1j!(k-j)!f1jf¯1k-j,
where k=2,3,…,n. Then
(83)f*Z=U+V¯,
where
(84)U=∑k=1n‍(Zfk)∂∂ωk,V¯=∑k=1n‍(Zf¯k)∂∂ω¯k.
In fact, for anyφ∈Ck(Sn),
(85)(f*Z)φ=Z(φ∘f)=Z[φ(f1,f2,…,fn,f¯1,f¯2,…,f¯n)]=∑k=1n‍∂φ∂ωkZfk+∑k=1n‍∂φ∂ω¯kZf¯k=[∑k=1n‍(Zfk)∂∂ωk+∑k=1n‍(Zf¯k)∂∂ω¯k]φ.
Letq=T-1(p)∈Sn; by (30), (34), (72), and (80) together with the left-invariant property of ZG,
(86)f*q(Z(q))=(T-1∘f~∘T)*q(Z(q))=T*-1∘f~*∘T*q(Z(q))=T*-1∘f~*p(ZG(p))=T*-1∘τf~(p)*0∘[Df~*(p)ZG(0)]=T*f~(p)-1∘τf~(p)*0×(c2+λ2ce-iθZG(0)+c2-λ2ceiθZ¯G(0))=T*f~(p)-1(c2+λ2ce-iθZG(f~(p))+c2-λ2ceiθZ¯G(f~(p)))=c2+λ2ce-iθZ(f(q))+c2-λ2ceiθZ¯(f(q)).
Since f is P-differentiable a.e., we have
(87)f*Z=c2+λ2ce-iθZ+c2-λ2ceiθZ¯,a.e.
Consequently, U,V¯ in (83) have the following forms
(88)U=∑k=1n‍(Zfk)∂∂ωk=c2+λ2ce-iθZ,a.e.,V¯=∑k=1n‍(Zf¯k)∂∂ω¯k=c2-λ2ceiθZ¯,a.e.
Hence, if λ=c2, then Z¯fk=0, k=1,2,…,n; if λ=-c2, then Zfk=0, k=1,2,…,n. Since Z and f are smooth, Theorem 1 follows.Remark 9.
The CR automorphism ofGn(n>3) is unknown now. It is also very interesting to determine it, since if this can be solved, we can get the 1-quasiconformal mappings on these groups.



---

*Source: 930571-2014-04-24.xml*