[Paper Review] Regularity And Extremality Of Quasiconformal Homeomorphisms On CR 3-Manifolds

Puqi Tang|arXiv (Cornell University)|Dec 30, 1993

Holomorphic and Operator Theory12 references16 citations

TL;DR

This paper establishes the regularity and extremality of quasiconformal homeomorphisms on 3-dimensional CR manifolds, particularly CR circle bundles over flat tori. Using modulus of curve families and sub-Riemannian geometry, it constructs extremal mappings that stretch and compress along Legendrian foliations—mirroring Teichmüller mappings on Riemann surfaces—proving they minimize dilatation in given homotopy classes.

ABSTRACT

This paper first studies the regularity of conformal homeomorphisms on smooth locally embeddable strongly pseudoconvex CR 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 behaviors to Teichmüller mappings on Riemann surfaces.

Motivation & Objective

• To establish the regularity of conformal homeomorphisms on smooth, strongly pseudoconvex, locally embeddable CR 3-manifolds with $L^1_{\text{loc}}$ horizontal derivatives.
• To define quasiconformality globally via modulus of curve families, linking infinitesimal distortion to global geometric behavior.
• To construct extremal quasiconformal homeomorphisms in specific homotopy classes on CR circle bundles over flat tori.
• To demonstrate that these extremal mappings preserve transverse Legendrian foliations and are equivariant under the circle action.
• To generalize Teichmüller-type extremality results from Riemann surfaces to the CR setting using length-area and modulus arguments.

Proposed method

• Uses the analytic definition of quasiconformality from [12], requiring ACL regularity and bounded distortion of horizontal derivatives.
• Applies the modulus of curve families as a global measure of quasiconformal distortion, proving a $C^2$ diffeomorphism is quasiconformal iff it distorts moduli by a bounded factor.
• Constructs extremal mappings by analyzing the modulus of vertical curve families $\Gamma_a$ parameterized by integral curves of a vector field $X$.
• Employs a length-area argument via the inequality $\text{Mod}_{M_1}(\Gamma_a) \leq (1/(2a))^4 \text{vol}(M_1)$, derived from $L^4$-norm estimates on curves.
• Uses homotopy lifting and rectifiable curve minimization to bound the length of image curves under $f$, leading to an upper bound on $\text{Mod}_{M_2}(f(\Gamma_a))$.
• Applies Lemma 4.5 to show that the $L^4$-norm of a form along fibers is independent of parameter $s$, enabling volume integral simplification.

Experimental results

Research questions

• RQ1 Under what regularity conditions is a conformal homeomorphism between smooth, strongly pseudoconvex CR 3-manifolds necessarily smooth and CR?
• RQ2 Can the modulus of curve families be used to characterize quasiconformal mappings globally on CR 3-manifolds?
• RQ3 Do extremal quasiconformal homeomorphisms exist in given homotopy classes on CR circle bundles over flat tori?
• RQ4 How do extremal mappings behave with respect to Legendrian foliations and the circle action in the CR setting?
• RQ5 To what extent do CR extremal mappings resemble Teichmüller mappings on Riemann surfaces in structure and distortion properties?

Key findings

• A conformal homeomorphism between smooth, strongly pseudoconvex, locally embeddable CR 3-manifolds with $L^1_{\text{loc}}$ horizontal derivatives is necessarily smooth and CR.
• A $C^2$ diffeomorphism is quasiconformal if and only if it distorts the modulus of certain curve families by a bounded factor, establishing a global characterization.
• For CR circle bundles over flat tori, extremal quasiconformal homeomorphisms exist in specified homotopy classes and preserve two transverse Legendrian foliations.
• These extremal mappings stretch along one Legendrian foliation by a factor of $\sqrt{K}$ and compress along the other by the same factor, mimicking Teichmüller maps.
• The extremal mapping $f_0$ is equivariant under the circle action in the transversal direction, as it acts as the identity on the circle fibers.
• By taking $a \to \infty$, the inequality $(\sqrt{K} - A/a)^2 \leq K(f)$ implies $K \leq K(f)$, proving that the maximal dilatation is minimized for the constructed extremal map.