Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 21, 1996, 289–308

REGULARITY AND EXTREMALITY OF QUASICONFORMAL HOMEOMORPHISMS ON CR 3-MANIFOLDS

Puqi Tang

Purdue University, Department of Mathematics West Lafayette, IN 47907, USA; tang@math.purdue.edu

Abstract. This paper first studies the regularity of conformal homeomorphisms on smooth locally embeddable strongly pseudoconvex Cauchy–Riemann manifolds. Then moduli of curve families are used to estimate the maximal dilatations of quasiconformal homeomorphisms. On certain CR 3-manifolds, namely, CR circle bundles over flat tori, extremal quasiconformal homeomorphisms in some homotopy classes are constructed. These extremal mappings have similar behavior to Teichmüller mappings on Riemann surfaces.

1. Introduction

A contact manifold M is a manifold of odd dimension with a nowhere integrable distribution HM of tangent hyperplanes. A Cauchy–Riemann (CR) manifold is a contact manifold M endowed a complex structure on the contact bundle HM. Two CR manifolds are equivalent if there is a diffeomorphism between them which respects both contact and CR structures. Generally, between any two CR structures assigned on the same contact manifold, there may be no such so-called CR diffeomorphism between them. Therefore we consider those diffeomorphisms between CR manifolds which preserve the underlying contact structures and distort the CR structures in a bounded fashion. They are called quasiconformal diffeomorphisms. Quasiconformality can be generalized to certain homeomorphisms with much weaker regularity. In a class of homeomorphisms between two CR manifolds, an extremal mapping is a quasiconformal homeomorphism which distorts the CR structures in a minimal way. This paper studies regularity of quasiconformal homeomorphisms and extremal quasiconformal homeomorphisms on smooth strongly pseudoconvex CR manifolds.

The notion of quasiconformal homeomorphisms is a new tool to study CR structures as initiated by Korányi and Reimann. In this paper, we use an analytic definition of quasiconformal homeomorphisms given in [13] which is a generalization of the one given by Korányi and Reimann in [5]. We restrict ourselves to the 3-dimensional case here not only because the notion of quasiconformality is not invariant under CR transformations in higher dimensional cases, but also because 3-dimensional CR structures are among the most interesting objects in the theory of CR manifolds. We refer to [7] for details about the second point.

1991 Mathematics Subject Classification: Primary 32G07.

The purpose of defining quasiconformal homeomorphisms is to introduce variants of CR diffeomorphisms. But quasiconformal homeomorphisms are variants of conformal homeomorphisms by default. Hence it is important to know if conformal homeomorphisms are actually CR diffeomorphisms. Korányi and Reimann proved that C^{4} conformal homeomorphisms on Heisenberg groups must be smooth and CR ([6, Theorem 8]). By applying a regularity theorem of weak CR mappings of Pinchuk and Tsyganov, we generalize Korányi and Reimann’s result to that a conformal homeomorphism f between two smooth, strongly pseudoconvex, locally embeddable CR manifolds must be smooth and CR, if f has L_{loc}^{1} horizontal derivatives (Theorem 2.3).

To study the extremality of quasiconformal homeomorphisms is a global problem. But our analytic definition of quasiconformality is proposed infinitesimally. Therefore we need some global notion to describe the quasiconformality. The one best fitting our later developments is the notion of moduli of curve families. We prove that a C^{2} diffeomorphism is quasiconformal if and only if it preserves moduli of certain curve families up to a fixed bounded multiple (Theorem 3.3). On the other hand, a homeomorphism satisfying this property is absolutely continuous on lines (ACL) (Theorem 3.4) and has L_{loc}^{4} horizontal derivatives (Corollary 3.5).

Between CR circle bundles over flat tori, we construct extremal quasiconformal homeomorphisms in certain homotopy classes (Theorem 4.2). There are two transversal Legendrian foliations such that the extremal homeomorphism constructed preserves these two foliations. More precisely, it is a stretching by a constant factor along leaves of one foliation and a compressing by the same factor along leaves of another foliation. This behavior is analogous to that of Teichmüller mappings on Riemann surfaces (see [14] or [1]). The generator T of the circle action is transversal to the contact bundle. In this transversal direction, the extremal mappings are equivariant under the circle action.

But on an arbitrary CR 3-manifold with a transversal free circle action, an extremal quasiconformal homeomorphism is not necessarily equivariant under the circle action. Such CR manifolds are constructed in [13] so that no extremal quasiconformal mapping between them is equivariant.

This work is heavily influenced by the theory of quasiconformal homeomorphisms on Riemann surfaces and Teichmüller theory. Numerous proofs in this paper are motivated by the proofs of the analogous facts on Riemann surfaces. For example, the construction of the extremal homeomorphisms made in Theorem 4.2, one of the main results of this paper, can find its root in Grötzsch’s theorem which is proved by a length-area argument [9]. Teichmüller generalized this result to closed Riemann surfaces, in particular tori, by an ergodic version of the length-area argument [14]. The notion of modulus of a curve family is a formalism of the length-area (volume) argument. In Section 4, we reformulate Teichmüller’s method to the CR setting by computing moduli of some special families of curves and successfully find the extremal quasiconformal homeomorphisms in certain homotopy classes of mappings between CR circle bundles over flat tori.

Acknowledgement. This author is very grateful to his academic advisor, László Lempert for the guidance with great insights. Thanks also to David Drasin and Juha Heinonen for helpful talks.

2. Regularity of conformal homeomorphisms

For j = 1, 2, let M_{j} be a smooth strongly pseudoconvex CR 3-manifolds. The contact bundle HM_{j} is assumed to be smooth and orientable, that is, there exists smooth global 1-form η_{j} on M_{j}, which is called a contact form, so that HM_{j} = Ker η_{j}. Let J_{j}: HM_{j} → HM_{j} denote the CR structure on M_{j}. H^{0,1}M_{j} ≜ {X+iJ_{j}X | X ∈ HM_{j}} ⊂ C⊗HM_{j} is the (0, 1) tangent bundle on M_{j}. ∧^{1,0}M_{j} ≜ {linear functionals ψ: C ⊗ HM_{j} → C | ψ(J_{j}X) = iψ(X), for X ∈ HM_{j}}. For a non-zero X ∈ HM_{j}, {X, J_{j}X} determines an orientation of HM_{j}.

A continuous mapping f: M_{1} → M_{2} is said to be absolutely continuous on lines (ACL) if for any open set with a smooth contact fibration, f is absolutely continuous along all fibers in this fibration except a subfamily of measure zero. Here a subfamily of fibers of the fibration is said to have measure zero if intersections of these fibers with any transversal smooth surface has measure zero on the surface (see [12] for more details).

Definition 2.1. A homeomorphism f: M_{1} → M_{2} is said to be K -quasicon-formal for a finite constant K ≥ 1 if

(i) f is ACL;

(ii) f is differentiable almost everywhere, and its differential f_{∗} preserves the contact structures, i.e., f_{∗}(H_{q}M_{1}) ⊂ H_{f(q)}M_{2} , for almost every q ∈ M_{1} .

(iii) for norms | · |_{1} and | · |_{2} defined by any Hermitian metrics on HM_{1} and HM_{2} respectively,

(2.1)  K(f)(q) = \frac{max_{X∈H_{q}M_{1},|X|_{1}=1} |f_{∗}(X)|_{2}}{min_{X∈H_{q}M_{1},|X|_{1}=1} |f_{∗}(X)|_{2}} ≤ K < ∞,

for almost all q ∈ M_{1} . The number K(f) = ess sup_{q∈M_{1}} K(f)(q) is called the maximal dilatation of f. f is conformal if K(f) = 1. K(f) = ∞ if f is not K -quasiconformal for any finite K ≥ 1.

It is easy to see that K(f)(q) does not depend on the choice of the Hermitian norms | · |_{1} and | · |_{2} . This is not true if the underlying CR manifolds have dimensions > 3.

A mapping f: M_{1} → M_{2} is said to have L_{loc}^{p} horizontal derivatives for p ≥ 1 if for any smooth function h: M_{2} → R and any smooth local section X of HM_{1} on an open set U ⊂⊂ M_{1} , the function h ∘ f has derivative at almost all points along the trajectory of X passing through q for almost all q ∈ U, and the derivative X(h ∘ f) is in L^{p}(U).

A mapping f: M_{1} → M_{2} is said to have L_{loc}^{p} weak horizontal derivatives for some p ≥ 1 if for any smooth function h: M_{2} → R and any open set U ⊂⊂ M_{1} with a smooth local section X of HM_{1} on U, there exists a function g ∈ L_{loc}^{p}(M_{1}) so that

(2.2)  ∫_{U} Xϕ · (h ∘ f) dv_{1} = − ∫_{U} g ϕ dv_{1}

for all ϕ ∈ C_{0}^{∞}(U), where dv_{1} is a smooth volume form on M_{1} . Certainly, the function g depends on the choice of dv_{1} .

For the proof of the next theorem, we fix a norm | · | on HM_{1} . A C^{1} curve on a contact manifold is called Legendrian if it is tangent to the contact structure. For any Legendrian curve γ: I → M_{1} with an interval I ⊂ R and a function g defined on an open neighborhood of γ, define the line integral

∫_{γ} g = ∫_{I} g(γ(t))|γ′(t)| dt.

Theorem 2.2. A mapping f: M_{1} → M_{2} is ACL and has L_{loc}^{p} horizontal derivatives for some p ≥ 1 if and only if f has L_{loc}^{p} weak horizontal derivatives.

Proof. First assume that the homeomorphism f: M_{1} → M_{2} is ACL and has L_{loc}^{p} horizontal derivatives. Let h: M_{2} → R be a smooth function and U ⊂⊂ M_{1} be any open set with a smooth section X ≠ 0 of HM_{1} on it.

We can assume that the trajectories Γ = {γ} of X form a contact fibration of U by shrinking U appropriately. Let Γ_{1} be the subfamily of those γ ∈ Γ along which f is absolutely continuous. Then Γ \ Γ_{1} has measure zero. Along γ ∈ Γ_{1} , h ∘ f is absolutely continuous, so X(h ∘ f) exists almost everywhere on γ and

(2.3) ∫_{γ} Xϕ · (h ∘ f) + ∫_{γ} ϕ · X(h ∘ f) = ∫_{γ} X(ϕ · h ∘ f) = 0,  ∀ ϕ ∈ C_{0}^{∞}(U).

We have topological and differential structures on Γ such that the natural projection p: U → Γ is open and smooth. Then Γ becomes a smooth surface. Let t be the parameter (with respect to a local coordinate system on U) of the flow generated by X and ω be any area form on Γ. Then dv_{1} ≜ p^{∗}ω ∧ dt is a volume form of U. Integrating the expressions in (2.3) against ω with respect to γ ∈ Γ_{1} , by the ACL property of f, local L^{p} integrability of X(h∘f), and Fubini’s theorem, we obtain

(2.4)  ∫_{U} Xϕ · (h ∘ f) dv_{1} = − ∫_{U} ϕ · X(h ∘ f) dv_{1}.

So the weak derivative of h ∘ f in the X direction is given by X(h ∘ f) ∈ L_{loc}^{p}.

Conversely, assume f has L_{loc}^{p} weak horizontal derivatives. For any open set U ⊂⊂ M_{1} and a smooth contact fibration Γ of U, let X be the nonzero horizontal vector field on U so that X_{q}, for any q ∈ U, is the tangent vector at q of the fiber γ ∈ Γ passing through q. Let B ⊂⊂ U be an open set with coordinate system {(x, y, t) | a_{1} < x < a_{2}, b_{1} < y < b_{2}, c_{1} < t < c_{2}}, such that X = ∂/∂t.

For any smooth function h on M_{1} , h ∘ f has L^{p}(U) weak derivative in the direction X. Denote it by ψ ∈ L^{p}(U). Hence there exists a sequence of C^{1} functions g_{n} on B such that g_{n} converges to h ∘ f uniformly in B and Xg_{n} converges to ψ in L^{p}(B). Let B_{x,y,t} = (a_{1}, x) × (b_{1}, y) × (c_{1}, t), R_{x,y} = (a_{1}, x) × (b_{1}, y).

(2.5) ∫_{B_{x,y,t}} Xg_{n}(u, v, w) du dv dw = ∫_{R_{x,y}}(g_{n}(u, v, t) − g_{n}(u, v, c_{1})) du dv.

Hence by taking limits, we have

(2.6) ∫_{B_{x,y,t}} ψ(u, v, w) du dv dw = ∫_{R_{x,y}}((h ∘ f)(u, v, t) − (h ∘ f)(u, v, c_{1})) du dv.

Let {t_{n}} be a countable dense set of (c_{1}, c_{2}). (2.6) implies for each t_{n}, there exists a set E_{n} ⊂ R_{a_{2},b_{2}} such that R_{a_{2},b_{2}} \ E_{n} is of measure zero and

(2.7) ∫_{c_{1}}^{t_{n}} ψ(x, y, w) dw = (h ∘ f)(x, y, t_{n}) − (h ∘ f)(x, y, c_{1}),  ∀ (x, y) ∈ E_{n}.

Then (2.7) is true for all (x, y) ∈ E ≜ ∩E_{n} and all t_{n}. Hence by continuity of both sides in t,

(2.8) ∫_{c_{1}}^{t} ψ(x, y, w) dw = (h∘f)(x, y, t)−(h∘f)(x, y, c_{1}), ∀ (x, y) ∈ E, t ∈ (c_{1}, c_{2}).

So (h ∘ f)(x, y, t) is absolutely continuous in t for (x, y) ∈ E. Note R_{a_{2},b_{2}} \ E has measure zero. So f is ACL since B ⊂⊂ U is an arbitrary rectangular coordinate chart and U ⊂⊂ M_{1} is arbitrary in M_{1} . Moreover, X(h ∘ f) exists almost everywhere and X(h ∘ f) = ψ ∈ L^{p}(U) on U. So f has L_{loc}^{p} horizontal derivatives.

Remark An equivalent result on Heisenberg groups was proved by Korányi and Reimann ([4, Proposition 9]). Our proof of the second part is different from theirs.

When M_{2} is locally embeddable into some Euclidean space C^{k}, a homeomorphism f: M_{1} → M_{2} is said to be weakly CR if ∫_{U}(h ∘ f) · \overline{Z}ϕ = 0 for any smooth CR function h: M_{2} → C, open set U ⊂⊂ M_{1} , ϕ ∈ C_{0}^{∞}(U) and \overline{Z} ∈ H^{0,1}M_{1} . Here h is said to be CR if \overline{Z}f = 0 for all \overline{Z} ∈ H^{0,1}M_{1} .

Theorem 2.3. Let M_{1} and M_{2} be two smooth, strongly pseudoconvex, locally embeddable CR 3-manifolds. Assume f: M_{1} → M_{2} is a conformal homeomorphism with L_{loc}^{1} horizontal derivatives and at almost all points of differentiability f_{∗} preserves the orientation of the contact bundles determined by CR structures. Then f is smooth and CR.

Proof. A simple linear algebra argument shows that at almost all points q ∈ M_{1} where f is differentiable

(2.9)  K(f)(q) = \frac{1 + |μ(q)|}{1 − |μ(q)|}  with  |μ(q)| = |\frac{⟨f^{∗}ψ_{2}, \overline{Z} ⟩}{⟨f^{∗}ψ_{2}, Z⟩}|(q),

for any nonzero \overline{Z} ∈ H^{0,1}M_{1} and nonzero ψ_{2} ∈ ∧^{1,0}M_{2} .

Thus K(f) = 1 implies that for almost every q ∈ M_{1} , \overline{Z} ∈ H_{q}^{0,1}M_{1} and ψ_{2} ∈ ∧_{f(q)}^{1,0}M_{2} , we have ⟨f^{∗}ψ_{2}, \overline{Z} ⟩ = 0. Hence for any CR function h on M_{2} ,

(2.10)  \overline{Z}(h ∘ f) = ⟨d(h ∘ f), \overline{Z} ⟩ = ⟨f^{∗} dh, \overline{Z} ⟩ = 0,

since dh|_{C⊗HM} ∈ ∧^{1,0}M_{2} . Theorem 2.2 says that f is weakly CR. M_{1} and M_{2} are locally embeddable implies they are locally embeddable into C^{2} as hypersurfaces. A theorem of Pinchuk and Tsyganov ([10, Theorem 2]) asserts that such f must be smooth, hence CR.

3. Moduli of curve families

Let M be a smooth, compact, contact 3-manifold. We always assume HM is smooth and oriented. A sub-Riemannian metric on M with respect to HM is a smooth positive definite quadratic form on HM. Fix a sub-Riemannian metric on M with respect to HM momentarily, and denote by | · | the corresponding norm on HM. For the general theory of sub-Riemannian geometry, we refer to [12] and [2].

The sub-Riemannian metric on M can be extended to a Riemannian metric on M canonically as follows. Let ω be the oriented area form on HM corresponding to the sub-Riemannian metric. There exists a unique contact form η so that dη|_{HM} = ω. Let T be the characteristic vector field of η, namely, T is the unique vector field satisfying that T ⌟η = 1 and T ⌟dη = 0. Declaring T a unit vector orthogonal to HM, we obtain a Riemannian metric which is called the canonical extension of the sub-Riemannian metric. The positive volume form of this Riemannian metric is dv ≜ dη ∧ η.

A curve γ: I_{γ} → M with an interval I_{γ} ⊂ R is called locally rectifiable if γ is absolutely continuous and γ′(t) is tangent to HM for almost all t ∈ I_{γ}. γ is called rectifiable if γ is locally rectifiable and the length

(3.1)  l(γ) ≜ ∫_{I_{γ}} |γ′(t)| dt < ∞.

We set l(γ) = ∞ if γ is not rectifiable. For a locally rectifiable curve γ and a non-negative Borel-measurable function σ on M, define the line integral

(3.2)  ∫_{γ} σ = ∫_{I_{γ}} σ(γ(t))|γ′(t)| dt.

Definition 3.1. Let Γ be a family of curves γ: I_{γ} → M. An admissible measure for Γ is a Borel-measurable function σ: M → R such that σ ≥ 0 and ∫_{γ} σ ≥ 1, for all locally rectifiable γ ∈ Γ. Denote the set of admissible measures for Γ by A(Γ). The modulus of Γ is defined by

(3.3)  Mod_{M}(Γ) = inf_{σ∈A(Γ)} ∫_{M} σ^{4} dv.

Remark. (1) It is easy to see that if two sub-Riemannian metrics on M with respect to HM define the same conformal structure on HM, then they give the same value for Mod_{M}(Γ).

(2) If Γ_{r} ⊂ Γ consists of all locally rectifiable curves of Γ, then Mod_{M}(Γ_{r}) = Mod_{M}(Γ).

The following proposition shows that modulus, regarded as a measure of (locally rectifiable) curve families, generalizes the concept of measure zero used in the definition of ACL property. Henceforth if a property holds for all curves in a family Γ except a subfamily with zero modulus, we say this property is true for almost all curves in Γ.

Proposition 3.2. Let U be an open set of M, Γ a contact fibration of U, Γ_{1} ⊂ Γ. Then Γ_{1} has measure zero if and only if Mod_{M}(Γ_{1}) = 0.

Proof. Without loss of generality, we assume U is a domain of the coordinate system {(x, y, t) | a_{1} < x < a_{2}, b_{1} < y < b_{2}, c_{1} < t < c_{2}} and X = ∂/∂t is tangent to Γ. Let E ⊂ (a_{1}, a_{2}) × (b_{1}, b_{2}) so that Γ_{1} = {curves t ↦ (x, y, t) | (x, y) ∈ E}.

If Γ_{1} has measure zero, then ∫_{E} dx dy = 0. Notice

(3.4)  σ_{0} = \{ \begin{matrix}1/(c_{2} − c_{1}), & when (x, y) ∈ E, t ∈ (c_{1}, c_{2}), \\ 0, & otherwise, \end{matrix}

is an admissible measure for Γ_{1} . Therefore

(3.5)  Mod_{M}(Γ_{1}) ≤ c ∫_{U} σ_{0}^{4} dx dy dt = 0,

where c is a constant upper bound of the Jacobian J on U with J dx dy dt = dv. If Mod_{M}(Γ_{1}) = 0, then for any σ ∈ A(Γ_{1}) with σ = 0 outside U, γ ∈ Γ_{1} ,

(3.6)  1 ≤ (∫_{γ} σ)^{4} ≤ (∫_{γ} σ^{4})(∫_{γ} 1)^{3} = (c_{2} − c_{1})^{3} ∫_{γ} σ^{4}.

Taking the integral over E,

(3.7)  ∫_{E} dx dy ≤ c′(c_{2} − c_{1})^{3} ∫_{M} σ^{4}.

Hence

(3.8)  ∫_{E} dx dy ≤ c′(c_{2} − c_{1})^{3}Mod_{M}(Γ_{1}) = 0,

that is, Γ_{1} has measure zero.

Let M_{1} and M_{2} be two compact, smooth, strongly pseudoconvex CR 3manifolds with smooth contact forms η_{1} and η_{2} respectively. Here the roles of sub-Riemannian metrics on HM_{1} and HM_{2} are played by Hermitian metrics with respect to the CR structures J_{1} and J_{2} respectively.

Theorem 3.3. A C^{2} homeomorphism f: M_{1} → M_{2} is K -quasiconformal for a constant K ≥ 1 if and only if for any family Γ of C^{1} Legendrian curves on M_{1}

(3.9)  \frac{1}{K^{2}} Mod_{M_{1}}(Γ) ≤ Mod_{M_{2}}(f(Γ)) ≤ K^{2}Mod_{M_{1}}(Γ),

where f(Γ) = {f(γ) | γ ∈ Γ}.

Proof. Assume f: M_{1} → M_{2} is K -quasiconformal. Then f is contact, i.e., for contact forms η_{1} and η_{2} on M_{1} and M_{2} respectively, f^{∗}η_{2} = λη_{1} with a C^{1} function λ on M_{1} . Then

(3.10)  f^{∗}(dη_{2}) = dλ ∧ η_{1} + λ dη_{1}.

Therefore

(3.11)  f^{∗}(dη_{2}|_{HM_{2}}) = λ dη_{1}|_{HM_{1}}.

For j = 1, 2, the Levi form L_{j} on M_{j} is a symmetric bilinear form on HM_{j} defined by

(3.12)  L_{j}(X, Y) = dη_{j}(X, J_{j}Y),  for X, Y ∈ HM_{j}.

By replacing η_{j} by −η_{j}, if necessary, we can always assume that L_{j} is positive definite. Hence L_{j} is a Hermitian form on HM_{j}. With respect to the Levi forms on M_{1} and M_{2} , we define

(3.13)  \begin{aligned}λ_{1}(q) & = max_{Y ∈H_{q}M_{1}, |Y |_{1}=1} |f_{∗}(Y)|_{2}, \\ λ_{2}(q) & = min_{Y ∈H_{q}M_{1}, |Y |_{1}=1} |f_{∗}(Y)|_{2}. \end{aligned}

Then (3.11) implies that |λ| = λ_{1}λ_{2} . Moreover, (3.10) implies that

(3.14)  f^{∗}(dη_{2} ∧ η_{2}) = λ^{2} dη_{1} ∧ η_{1}.

Thus the Jacobian of f with respect to the volume forms dv_{1} = dη_{1} ∧ η_{1} on M_{1} and dv_{2} = dη_{2} ∧ η_{2} on M_{2} is J(f) = (λ_{1}λ_{2})^{2} .

For any non-negative Borel-measurable function σ_{2} ,

(3.15)  \begin{aligned}∫_{f(γ)} σ_{2} & = ∫_{I_{γ}} |f_{∗}(γ′(t))|_{2}σ_{2}(f(γ(t))) dt \\ & ≤ ∫_{I_{γ}} λ_{1}|γ′(t)|_{1}σ_{2}(f(γ(t))) dt = ∫_{γ} λ_{1} · σ_{2} ∘ f. \end{aligned}

Hence σ_{2} ∈ A(f(Γ)) implies that λ_{1} · σ_{2} ∘ f ∈ A(Γ).

On the other hand,

(3.16)  ∫_{f(γ)} σ_{2} ≥ ∫_{I_{γ}} λ_{2}|γ′(t)|_{1}σ_{2}(f(γ(t))) dt = ∫_{γ} λ_{2} · σ_{2} ∘ f.

Hence σ_{1} ∈ A(Γ) implies (σ_{1}/λ_{2}) ∘ f^{−1} ∈ A(f(Γ)). Therefore,

(3.17)

Mod_{M_{2}}(f(Γ)) = inf_{σ_{2}∈A(f(Γ))} ∫_{M_{2}} σ_{2}^{4} dv_{2} ≤ inf_{σ_{1}∈A(Γ)} ∫_{M_{2}}((\frac{σ_{1}}{λ_{2}} ) ∘ f^{−1})^{4} dv_{2}

= inf_{σ_{1}∈A(Γ)} ∫_{M_{1}} \frac{σ_{1}^{4}}{λ_{2}^{4}} · (λ_{1}λ_{2})^{2} dv_{1} ≤ K^{2} Mod_{M_{1}}(Γ).

(3.18)

Mod_{M_{1}}(Γ) = inf_{σ_{1}∈A(Γ)} ∫_{M_{1}} σ_{1}^{4} dv_{1} ≤ inf_{σ_{2}∈A(f(Γ))} ∫_{M_{1}}(λ_{1} · σ_{2} ∘ f)^{4} dv_{1}

= inf_{σ_{2}∈A(f(Γ))} ∫_{M_{2}} \frac{(λ_{1} ∘ f^{−1} · σ_{2})^{4}}{(λ_{1}λ_{2})^{2} ∘ f^{−1}} dv_{2} ≤ K^{2}Mod_{M_{2}}(f(Γ)).

Assume that f satisfies the inequalities (3.9). Then f must respect contact structure in the sense that f_{∗}(HM_{1}) ⊂ HM_{2} . Otherwise, there will be a point q ∈ M_{1} with a tangent vector X_{q} ∈ H_{q}M_{1} so that f_{∗}(X_{q}) ∉ H_{f(q)}M_{2} . f is C^{1} , so there is a neighborhood U of q with a nonzero smooth section X of HM_{1} on U so that f_{∗}(X_{q′}) is not contact for each q′ ∈ U. Let Γ be the family of trajectories of X, and shrink U appropriately so that Mod_{M_{1}}(Γ) ≠ 0. On the other hand, curves in f(Γ) are not Legendrian, hence not locally rectifiable. So Mod_{M_{2}}(f(Γ)) = 0 by Definition 3.1. Thus f cannot satisfy (3.9) for this family Γ. This contradiction shows that f must be contact.

If f is not K -quasiconformal, there exists an open set U ⊂ M_{1} such that λ_{1}/λ_{2} ≥ K + ε, for some ε > 0. Let X be the vector field on U so that |X|_{1} = 1, |f_{∗}(X)|_{2} = λ_{2} . Let Γ be the family of trajectories of X in U. Then for any function σ_{2} on f(U), non-negative Borel-measurable,

(3.19)  ∫_{f(γ)} σ_{2} = ∫_{γ} σ_{2} ∘ f · |f_{∗}(X)|_{2} = ∫_{γ} λ_{2} · σ_{2} ∘ f.

So σ_{2} ∈ A(f(Γ)) if and only if σ_{1} ≜ λ_{2} · σ_{2} ∘ f ∈ A(Γ). Hence

Mod_{M_{2}}(f(Γ)) = inf_{σ_{2}∈A(f(Γ))} ∫_{M_{2}} σ_{2}^{4} dv_{2}

= inf_{σ_{1}∈A(Γ)} ∫_{f(U)}((\frac{σ_{1}}{λ_{2}} ) ∘ f^{−1})^{4} dv_{2}

= inf_{σ_{1}∈A(Γ)} ∫_{U}(\frac{σ_{1}}{λ_{2}} )^{4} · (λ_{1}λ_{2})^{2} dv_{1} (3.20)

= inf_{σ_{1}∈A(Γ)} ∫_{U}(\frac{λ_{1}}{λ_{2}} )^{2}σ_{1}^{4} dv_{1} ≥ (K + ε)^{2} Mod_{M_{1}}(Γ).

Therefore, (3.9) implies that

(3.21)  (K + ε)^{2}Mod_{M_{1}}(Γ) ≤ K^{2}Mod_{M_{1}}(Γ).

But we can always choose U such that Mod_{M_{1}}(Γ) ≠ 0. So (3.9) cannot be true for such Γ. Hence λ_{1}/λ_{2} ≤ K on M_{1} , i.e., f is K -quasiconformal.

It would be important to know if we can use (3.9) to define quasiconformality for a homeomorphism f: M_{1} → M_{2} . The rest of this section is devoted to this problem.

Theorem 3.4. If f: M_{1} → M_{2} is a homeomorphism so that for a constant K ≥ 1 and any curve family Γ which forms a smooth contact fibration of an open set in M_{1}

(3.22)  Mod_{M_{1}}(Γ) ≤ K^{2} Mod_{M_{2}}(f(Γ)),

then f is ACL.

Proof. Let U ⊂ M_{1} be an open set with a smooth contact fibration Γ, X ≠ 0 be a horizontal vector field on U tangent to Γ. By replacing X by X/|X|_{1} , we can assume |X|_{1} = 1. Shrink U, if necessary, so that there is a smooth surface S ⊂ U which intersects each fiber of Γ transversally once and only once. Choose a local coordinate system such that fibers γ ∈ Γ is parametrized by t, X = ∂/∂t and γ(0) ∈ S. Let p: U → S denote the natural projection given by γ(t) ↦ γ(0).

Recall that Hermitian metrics on HM_{1} and HM_{2} can be extended canonically to Riemannian metrics on M_{1} and M_{2} respectively. Restricting the Riemannian metric on M_{1} to S makes S a Riemannian 2-manifold. Let ω be the area form on S, and A_{ω}(E) ≜ ∫_{E} ω the ω -area of a measurable set E ⊂ S. Define a set function F for measurable set E ⊂ S by letting F (E) = vol (f(p^{−1}(E))), where vol refers to the Riemannian volume on M_{2} . Then Lebesgue’s theorem ([15, Theorem 23.5]) asserts that F has finite derivatives at all points in S_{1} with respect to ω -area, for a subset S_{1} ⊂ S with A_{ω}(S \ S_{1}) = 0. We want to prove f is absolutely continuous along the fibers of Γ passing through points in S_{1} .

For q ∈ S_{1} , let D_{r} ⊂ S be a disc centered at q with radius r, γ_{q}: I → U be the fiber of γ passing through q. Take any sequence t′_{1}, t′′_{1}, t′_{2}, t′′_{2}, …, t′_{k}, t′′_{k} ∈ I such that

(3.23)  t′_{1} < t′′_{1} < t′_{2} < t′′_{2} ⋯ < t′_{k} < t′′_{k}.

Let Δt_{j} ≜ t′′_{j} − t′_{j}. When max_{1≤j≤k} Δt_{j} is small enough,

(3.24)  d_{r,j} ≤ Δt_{j} ≤ 2 d_{r,j},

where d_{r,j} is the sub-Riemannian distance between the set B′_{r,j} ≜ {γ(t′_{j}) | γ ∈ Γ, γ(0) ∈ D_{r}} and the set B′′_{r,j} ≜ {γ(t′′_{j}) | γ ∈ Γ, γ(0) ∈ D_{r}}. For j = 1, 2, …, k, denote

R_{r,j} ≜ {γ(t) | γ ∈ Γ, γ(0) ∈ D_{r}, t′_{j} ≤ t ≤ t′′_{j}}

and let

Γ_{r,j} ≜ {γ_{j} = γ|_{[t′_{j},t′′_{j}]} | γ ∈ Γ, γ(0) ∈ D_{r}},

a contact fibration of R_{r,j}. Then by (3.22),

(3.25)  Mod_{M_{1}}(Γ_{r,j}) ≤ K^{2} Mod_{M_{2}}(f(Γ_{r,j})).

Next we use length-volume argument to give an estimate for Mod_{M_{1}}(Γ_{r,j}). For σ ∈ A(Γ_{r,j}) and γ_{j} ∈ Γ_{r,j},

(3.26)  1 ≤ (∫_{γ_{j}} σ)^{4} ≤ (∫_{γ_{j}} σ^{4})(∫_{γ_{j}} 1)^{3} = (Δt_{j})^{3} ∫_{γ_{j}} σ^{4}.

Integrating each term against ω over D_{r}, then

(3.27)  A_{ω}(D_{r}) ≤ c(Δt_{j})^{3} ∫_{R_{r,j}} σ^{4} dv_{1}.

where c is constant, dv_{1} is the volume form of the Riemannian metric on M_{1} . By (3.3) and (3.24),

(3.28)  A_{ω}(D_{r}) ≤ 8cd_{r,j}^{3} Mod_{M_{1}}(Γ_{r,j}).

On the other hand, let δ_{r,j} be the sub-Riemannian distance between f(B′_{r,j}) and f(B′′_{r,j}). Then 1/δ_{r,j} ∈ A(f(Γ_{r,j})). Thus

(3.29)  Mod_{M_{2}}(f(Γ_{r,j})) ≤ \frac{1}{δ_{r,j}^{4}} vol (f(R_{r,j})).

Combining (3.25), (3.28) and (3.29), we have

(3.30)  (A_{ω}(D_{r}))^{1/4} δ_{r,j} ≤ (8cK^{2})^{1/4}d_{r,j}^{3/4}(vol (f(R_{r,j})))^{1/4}.

Summing (3.30) over j,

(3.31)

(A_{ω}(D_{r}))^{1/4} ∑_{j=1}^{k} δ_{r,j} ≤ (8cK^{2})^{1/4} ∑_{j=1}^{k} d_{r,j}^{3/4}(vol (f(R_{r,j})))^{1/4}

≤ (8cK^{2})^{1/4}(∑_{j=1}^{k} d_{r,j})^{3/4}(∑_{j=1}^{k} vol (f(R_{r,j})))^{1/4}

≤ (8cK^{2})^{1/4}(∑_{j=1}^{k} d_{r,j})^{3/4}(vol (f(p^{−1}(D_{r}))))^{1/4}.

So by (3.24),

(3.32)  ∑_{j=1}^{k} δ_{r,j} ≤ (8cK^{2})^{1/4} (∑_{j=1}^{k} Δt_{j})^{3/4}(\frac{F (D_{r})}{A_{ω}(D_{r})})^{1/4}.

Letting r → 0,

(3.33)  ∑_{j=1}^{k} δ_{j} ≤ (8cK^{2}F′(q))^{1/4}(∑_{j=1}^{k} Δt_{j})^{3/4},

where δ_{j} ≜ lim_{r→0} δ_{r,j}, i.e., the sub-Riemannian distance between f(γ_{q}(t_{j})) and f(γ_{q}(t_{j+1})). Hence f is absolutely continuous along γ_{q}, for q ∈ S_{1} . So f is ACL.

Corollary 3.5. If a homeomorphism f: M_{1} → M_{2} satisfies (3.22), then f has horizontal derivatives almost everywhere on M_{1} in the sense that for any open set U ⊂ M_{1} with a smooth section X of HM_{1} on U and any smooth function h on f(U), X(h∘f) exists almost everywhere on U. Moreover f has L_{loc}^{4} horizontal derivatives.

Proof. We need only prove the last statement for non-zero X. Using notations as in the proof of Theorem 3.4, we will show X(h ∘ f) ∈ L_{loc}^{4} along γ_{q} for q ∈ S where F′(q) exists. Without loss of generality, we assume γ_{q}: I = [a, b] → M_{1} is a closed curve. Let t_{0} = a, t_{1}, …, t_{N} = b be a partition of I. For j = 0, 1, …, N−1, define δ_{r,j}, δ_{j}, d_{r,j}, and d_{j} similarly. Then the same proof of (3.30) shows that

(3.34)  (\frac{δ_{r,j}}{d_{r,j}} )^{4}d_{r,j} ≤ 8cK^{2} \frac{vol (f(R_{r,j}))}{A_{w}(D_{r})},

when the partition of I is fine enough. Then

(3.35)  ∑_{j=1}^{N−1}(\frac{δ_{r,j}}{d_{r,j}} )^{4}d_{r,j} ≤ 8cK^{2} \frac{vol (f(p^{−1}(D_{r})))}{A_{w}(D_{r})}.

Letting r → 0, we obtain

(3.36)  ∑_{j=1}^{N−1}(\frac{δ_{j}}{d_{j}} )^{4}d_{j} ≤ 8cK^{2}F′(q).

Letting max_{0≤j≤N−1} |t_{j+1} −t_{j}| → 0, by an elementary argument based on Fatou’s lemma [11],

(3.37)  ∫_{γ_{q}} |f_{∗}(X)|^{4} ≤ 8cK^{2}F′(q).

Note F′ exists for almost all q ∈ D_{r} for a fixed small r > 0 and F′ is integrable over D_{r} by Lebesgue’s theorem. Taking integral of both sides of (3.37) over D_{r} against ω,

(3.38)  ∫_{p^{−1}(D_{r})} |f_{∗}(X)|^{4} dv_{1} ≤ 8c_{1}K^{2} ∫_{D_{r}} F′ω < ∞.

Hence f has L_{loc}^{4} horizontal derivatives.

4. Extremal quasiconformal homeomorphisms

In this section we will construct CR 3-manifolds M_{1} , M_{2} and a quasiconformal diffeomorphism f_{0}: M_{1} → M_{2} such that K(f_{0}) ≤ K(f), for any C^{2} quasiconformal homeomorphism f: M_{1} → M_{2} homotopic to f_{0} . M_{1} to be constructed is the quotient of the 3-dimensional Heisenberg group by a lattice. So we start with the Heisenberg group.

The 3-dimensional Heisenberg group H^{3} is the space R^{3} endowed with the group structure defined by

(4.1)  (x, y, t) (u, v, s) = (x + u, y + v, t + s + 2yu − 2xv).

The standard contact structure on H^{3} is given by the contact form

(4.2)  η̃ = −\frac{1}{2}y dx + \frac{1}{2}x dy + \frac{1}{4} dt.

The contact bundle HH^{3} = Ker η̃ has two global sections

(4.3)  X̃ = \frac{∂}{∂x} + 2y \frac{∂}{∂t},  Ỹ = \frac{∂}{∂y} − 2x\frac{∂}{∂t}

which span HH^{3} everywhere and are invariant under the left group translation. The standard CR structure on H^{3} is

(4.4)  J̃: HH^{3} → HH^{3},  X̃ ↦ Ỹ,  Ỹ ↦ −X̃.

Note that dη̃ = dx ∧ dy, thus the sub-Riemannian metric on HH^{3} determined by the area form dη̃ and the complex structure J̃, i.e., the Levi form, is the one making {X̃, Ỹ} orthonormal.

Next we study the geodesics on H^{3} with respect to this sub-Riemannian metric. Since the metric is invariant under left group translations, it suffices to study geodesics joining the origin O = (0, 0, 0) and a generic point q. Denote by p the projection from H^{3} to the horizontal plane P ≜ {t = 0}. It is easy to see that if γ is a rectifiable curve in H^{3} with respect to the sub-Riemannian metric, p(γ) is rectifiable with respect to the Euclidean metric on {t = 0}, and the respective lengths of γ and p(γ) coincide. The characterization of the geodesics given in the next theorem was first given by Korányi by studying the Euler–Lagrange equations [3]. The following proof, which is due to Lempert [8], is a geometric one.

Proposition 4.1. On H^{3} , any minimal geodesic connecting the origin O and q ∈ H^{3} is either a straight line segment if q ∈ P, or an arc of a helix whose projection to P is a circle if q ∉ P.

Proof. The conclusion is obvious if q ∈ P. Next we assume q ∉ P. We first consider the case p(q) = O, i.e., q = (0, 0, t) for some t ≠ 0. Let

(4.5)  γ: [0, l] → H^{3},  s ↦ (x(s), y(s), t(s))

be any oriented rectifiable curve joining O and q. Then

(4.6)  −2yx′ + 2xy′ + t′ = 0,

almost everywhere on γ by (4.2). Thus

(4.7)  t = ∫_{0}^{l} t′ ds = ∫_{p(γ)} 2y dx − 2x dy = −4 ∫_{Ω} dx ∧ dy,

where Ω is the 2-chain on P so that ∂Ω = p(γ). Now the sub-Riemannian length of γ is equal to the Euclidean length of p(γ). Obviously, the length of p(γ) is the minimal only when Ω is a simply connected domain with fixed area \frac{1}{4}|t|. Furthermore, p(γ) must be a circle if γ is a length minimizing geodesic joining O and q, by the isoperimetric property on the Euclidean plane.

If p(q) ≠ O, the above reasoning with some slight modifications can be applied. For instance, Ω is a 2-chain bounded by the union of p(γ) and the straight line segment from p(q) to O.

Take σ, τ ∈ R, τ ≠ 0. Let A = (1, 0, 0), B = (σ, τ, 0). Let Γ be the discrete subgroup of H generated by {A, B}. Left group translations by elements of Γ define a Γ-action. The CR structure on H is invariant under this Γ-action. We consider the quotient space M_{1} ≜ H/Γ. Let π: H → M_{1} denote the natural projection. M_{1} has a contact structure with the contact form η satisfying π^{∗}η = η̃ and a CR structure J_{1} inherited from J̃. Hence M_{1} becomes a CR manifold. Let X = π_{∗}X̃, Y = π_{∗}Ỹ. Then with respect to the sub-Riemannian metric uniquely determined by η and J_{1} , \{X, Y \} is orthonormal. Note also \overline{Z}_{1} = \frac{1}{2}(X + iY) ∈ T^{0,1}M_{1} . There is an R-action on H defined by

(4.8)  (x, y, t) ↦ (x, y, t + t′),  for t′ ∈ R.

This R-action is free and preserves the CR structures on H^{3} . The plane P can be regarded as the quotient space of this R-action. The natural projection p: H^{3} → P is isometric between the sub-Riemannian metric on H^{3} and the Euclidean metric on P. The free R-action on H^{3} commutes with the Γ-action. Thus it induces a free S^{1} -action on M_{1} so that M_{1} becomes a circle bundle over the torus Σ_{1} ≜ π(P). The torus Σ_{1} has a flat Riemannian metric inherited from the Euclidean metric on P. Denote the natural projection by π_{1}: M_{1} → Σ_{1} . Note the S^{1} -action preserves the CR structure on M_{1} . We call such a bundle M_{1} →^{π_{1}} Σ_{1} a CR circle bundle over a flat torus.

Given a constant K ≥ 1, define a new CR structure M_{2} on the contact structure of M_{1} by declaring that

(4.9)  \overline{Z}_{2} ≜ \frac{1}{2}(\sqrt{K} X + \frac{i}{\sqrt{K}} Y ) = \frac{K + 1}{4\sqrt{K}}(\overline{Z}_{1} + \frac{K − 1}{K + 1}Z_{1})

is an (0, 1) tangent vector. Obviously, the identity mapping f_{0}: M_{1} → M_{2} is quasiconformal and K(f_{0})(q) = K for all q ∈ M_{1} .

Theorem 4.2. Let f: M_{1} → M_{2} be a C^{2} homeomorphism homotopic to f_{0}: M_{1} → M_{2} . Then K(f) ≥ K.

Note the CR structure on M_{2} is also invariant under the circle action. So if S_{2} denotes the same smooth torus as S_{1} , but endowed with the complex structure induced from CR structure on M_{2} , then M_{2} →^{π_{1}} S_{2} is also a CR circle bundle.

Lemma 4.3 (Strichartz, see [12, Lemma 3.2]). For any q in a sub-Riemannian manifold M, there is an ε > 0 so that if q_{1}, q_{2} ∈ M with d(q_{1}, q) ≤ ε and d(q_{2}, q) ≤ ε, there exists a length minimizing curve joining q_{1} and q_{2} .

If α is a curve, denote by [α] the homotopy class with fixed end points of α. Recall that l(α) is the length of α with respect to the sub-Riemannian metric on M.

Lemma 4.4. Let α: [0, 1] → M be a curve on a compact sub-Riemannian manifold connecting two points q_{1} and q_{2} , inf l(β) be taken over all β ∈ [α]. Then this infimum is attained at a rectifiable curve α̃ ∈ [α].

Proof. Let ε_{q} > 0 be a number for q ∈ M determined by Lemma 4.3. Since M is compact, we can cover M with finitely many balls B(q_{j}, \frac{1}{3}ε_{q_{j}}), j = 1, 2, …, k with respect to the sub-Riemannian metric. Note any two points in the same B(q_{j}, \frac{1}{3}ε_{q_{j}}) can be joined by a length minimizing curve within B(q_{j}, ε_{q_{j}}).

Let L = inf l(β) over all β ∈ [α]. Let β_{n} ∈ [α] be curves such that lim_{n→∞} l(β_{n}) = L. Divide each β_{n} by points p_{n0} = q_{1}, p_{n1}, p_{n2}, …, p_{nN} = q_{2} ∈ β_{n} into subcurves τ_{n1}, τ_{n2}, …, τ_{nN} so that each τ_{nj} ⊂ B(q_{m_{nj}}, r) for some integer m_{nj} with 1 ≤ m_{nj} ≤ k. Here r denotes the radius \frac{1}{3}ε_{q_{m_{nj}}} for simplicity. Note this N is uniform for all n by appropriately choosing β_{n}.

Let σ_{nj} be a length minimizing curve in B(q_{m_{nj}}, 3r) to join the end points of τ_{nj} and σ_{n} = σ_{n1}σ_{n2} ⋯ σ_{nN}. Then since B(q_{m_{nj}}, 3r) is simply connected, σ_{nj} ∈ [τ_{nj}], and hence σ_{n} ∈ [β_{n}] = [α]. Furthermore

(4.10)  L ≤ l(σ_{n}) ≤ ∑_{j=1}^{N} l(σ_{nj}) ≤ ∑_{j=1}^{N} l(τ_{nj}) = l(β_{n}).

Therefore lim_{n→∞} l(σ_{n}) = L.

Since M is compact, there is a sequence {n_{j}} ⊂ N so that for each j = 1, 2, …, N −1, p_{nj} is convergent to some point p_{j}, when n_{j} → ∞. Note p_{j−1}, p_{j} ∈ B(q_{m_{j}}, r) for some integer m_{j} with 1 ≤ m_{j} ≤ k. Connect p_{j−1}, p_{j} by a length minimizing curve α_{j} ⊂ B(q_{m_{j}}, 3r) and let α̃ = α_{1}α_{2} ⋯ α_{N}. Then α̃ ∈ [α] and

(4.11)  l(α̃) = lim_{n_{i}→∞} l(σ_{n_{i}}) = L.

The next two lemmas will be crucial to establish the extremality of f_{0} described by Theorem 4.2.

Lemma 4.5. The flow of diffeomorphisms h_{t} generated by X on M_{1} preserves the volume form dv_{1} = dη ∧ η.

Proof. This is obvious since the the flow of diffeomorphisms h̃_{s} on H^{3} generated by X̃ is given by

(4.12)  h̃_{s}: (x, y, t) ↦ (x + s, y, t + 2sy),  s ∈ R

which preserves the volume form dη̃ ∧ η̃ = dx ∧ dy ∧ dt.

We define a sub-Riemannian metric on M_{2} such that \{\sqrt{K} X, Y/\sqrt{K} \} is orthonormal. Note this sub-Riemannian metric induces the same CR structure on M_{2} . Denote the corresponding curve length by l_{2} . On a sub-Riemannian manifold M, a geodesic γ: R → M is said to be (homotopically) conjugate point free if for any [a, b] ⊂ R the curve γ|_{[a,b]} is the unique length minimizing curve in its homotopy class of curves with end points γ(a) and γ(b).

Lemma 4.6. Integral curves of the vector field \sqrt{K} X are conjugate point free geodesics on M_{2} .

Proof. Lifting the sub-Riemannian metric on M_{2} to its universal covering H, we obtain a new sub-Riemannian metric on H. Let us call this new sub-Riemannian manifold H_{2} . On H_{2} there is a characterization of geodesics similar to Proposition 4.1.1. The trajectories of \sqrt{K} X̃ are exactly the straight geodesics on H_{2} . Indeed, the sub-Riemannian metric is invariant under the R-action (4.8), hence a rectifiable curve γ in H_{2} has the same length as p(γ) on P_{2} . Here P_{2} is the plane P endowed with the flat Riemannian metric chosen such that \sqrt{K} ∂/∂x, (1/\sqrt{K})(∂/∂y) are orthonormal. Therefore the x-axis, which is the trajectory of \sqrt{K} X̃ passing through O, is a conjugate point free geodesic. Note \sqrt{K} X̃ and the sub-Riemannian metric are invariant under the group left translation. Hence any other trajectory of \sqrt{K} X̃ is also a conjugate point free geodesic. The lemma follows from this fact.

Proof of Theorem 4.2. Let f: M_{1} → M_{2} be a C^{2} quasiconformal homeomorphism. For q ∈ M_{1} , let γ_{q,a}: [−a, a] → M_{1} be the integral curve of X so that γ_{q,a}(0) = q. Let Γ_{a} = {γ_{q,a} | q ∈ M_{1}}. Then by Theorem 3.4,

(4.13)  Mod_{M_{1}}(Γ_{a}) ≤ K(f)^{2} Mod_{M_{2}}(f(Γ_{a})).

First note the length of γ_{q,a} is 2a, so 1/2a ∈ A(Γ_{a}) and

(4.14)  Mod_{M_{1}}(Γ_{a}) ≤ ∫_{M_{1}}(\frac{1}{2a})^{4}dv_{1} = (\frac{1}{2a})^{4}vol (M_{1}).

For σ ∈ A(Γ_{a}),

(4.15)  1 ≤ (∫_{γ_{q,a}} σ)^{4} ≤ (∫_{γ_{q,a}} 1)^{3}(∫_{γ_{q,a}} σ^{4}) = (2a)^{3} ∫_{γ_{q,a}} σ^{4}.

That implies

(4.16)  \frac{1}{(2a)^{3}} ≤ ∫_{γ_{q,a}} σ^{4} = ∫_{−a}^{a}(σ(γ_{q,a}(s)))^{4} ds.

Integrating both sides of (4.16) against dv_{1} with respect to q over M_{1} ,

(4.17)  \begin{aligned}\frac{1}{(2a)^{3}} vol (M_{1}) & ≤ ∫_{M_{1}}(∫_{−a}^{a}(σ(γ_{q,a}(s)))^{4} ds) dv_{1} \\ & = ∫_{−a}^{a}(∫_{M_{1}}(σ(γ_{q,a}(s)))^{4} dv_{1}) ds \\ & = 2a ∫_{M_{1}} σ^{4} dv_{1}. \end{aligned}

The last equality in (4.17) is due to Lemma 4.5, which implies that

(4.18)  ∫_{M_{1}}(σ(γ_{q,a}(s)))^{4} dv_{1} = ∫_{M_{1}} σ^{4} h_{s}^{∗}(dv_{1}) = ∫_{M_{1}} σ^{4} dv_{1}

is independent of s. Therefore

(4.19)  (\frac{1}{2a})^{4}vol (M_{1}) ≤ ∫_{M_{1}} σ^{4} dv_{1},  ∀ σ ∈ A(Γ_{a}).

Combining (4.14) and (4.19), we get

(4.20)  Mod_{M_{1}}(Γ_{a}) = (\frac{1}{2a})^{4}vol (M_{1}).

Next we estimate Mod_{M_{2}}(f(Γ_{a})) in (4.13) for a quasiconformal homeomorphism f homotopic to f_{0}: M_{1} → M_{2} . Since f is homotopic to f_{0} , there is a continuous map H: [0, 1] × M_{1} → M_{2} so that

(4.21)  H(0, q) = f_{0}(q),  H(1, q) = f(q),  ∀ q ∈ M_{1}.

Let α_{q}: [0, 1] → M_{2} be a curve given by t ↦ H(t, q). Define G(q) = inf l_{2}(β) over all β ∈ [α_{q}]. Then by Lemma 4.4, G(q) is attained at a rectifiable curve α̃_{q} ∈ [α_{q}].

Now we prove that G(q) is continuous on M_{1} . When q_{1} is close enough to q on M_{1} , f_{0}(q_{1}) is close enough to f_{0}(q) so that there exist a length minimizing curve δ_{1} joining f_{0}(q), f_{0}(q_{1}) according to Lemma 4.3. Let δ_{2} = f(δ_{1}). Then

(4.22)  δ_{1}α̃_{q_{1}}δ_{2}^{−1} ∈ [α_{q}],  δ_{1}^{−1}α̃_{q}δ_{2} ∈ [α_{q_{1}}].

This implies

(4.23)  \begin{aligned}l_{2}(α̃_{q}) & ≤ l_{2}(δ_{1}) + l_{2}(α̃_{q_{1}}) + l_{2}(δ_{2}^{−1}), \\ l_{2}(α̃_{q_{1}}) & ≤ l_{2}(δ_{1}^{−1}) + l_{2}(α̃_{q}) + l_{2}(δ_{2}). \end{aligned}

In other words,

(4.24)  |G(q_{1}) − G(q)| ≤ l_{2}(δ_{1}) + l_{2}(δ_{2}).

Thus there exists A > 0 so that

(4.25)  G(q) ≤ A < ∞,  ∀ q ∈ M_{1},

since M_{1} is compact.

For a curve γ: [a, b] → M_{1} , α̃_{γ(a)}f(γ)α̃_{γ(b)}^{−1} ∈ [f_{0}(γ)]. By Lemma 4.6 and (4.25),

(4.26)  l_{2}(f_{0}(γ)) ≤ l_{2}(α̃_{γ(a)}) + l_{2}(f(γ)) + l_{2}(α̃_{γ(b)}^{−1}) ≤ 2A + l_{2}(f(γ)).

Applying (4.26) to γ = γ_{q,a} ∈ Γ_{a}, we obtain

(4.27)  2\sqrt{K} a ≤ 2A + l_{2}(f(γ_{q,a})).

Hence 1/(2\sqrt{K} a − 2A) ∈ A(f(Γ_{a})). So we have the following estimate:

(4.28) Mod_{M_{2}}(f(Γ_{a})) ≤ ∫_{M_{2}}(\frac{1}{2\sqrt{K} a − 2A})^{4} dv_{2} = (\frac{1}{2\sqrt{K} a − 2A})^{4}vol (M_{2}).

Note vol (M_{1}) = vol (M_{2}) ≠ 0, so by (4.13), (4.20) and (4.28), we have

(4.29)  (\sqrt{K} − \frac{A}{a})^{2} ≤ K(f).

Letting a → ∞, we get that K ≤ K(f).

Remark. In horizontal directions, the extremal quasiconformal homeomorphism f_{0} behaves as a stretching by the constant factor \sqrt{K} along trajectories of X and a compressing by the same factor along trajectories of JX. The generator T of the circle action is transversal to the contact bundle. In this transversal direction T, f_{0} is equivariant under the circle action since it is simply the identity mapping while M_{1} and M_{2} have the same circle action.

308 Puqi Tang

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Received 13 March 1995