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The affine ellipse map and its Beltrami coefficient

Statement

Assume the Axiom of Choice. Fix $μ \in C$ with $∣ μ ∣ < 1$ and define $f : C \to C$ by $f (z) = z + μ \overline{z}$ . Verify:

(a) $f$ is a homeomorphism with inverse $f^{- 1} (w) = \frac{w - μ \overline{w}}{1 - ∣ μ ∣^{2}},$ is orientation-preserving, and has $f_{z} = 1$ and $f_{\overline{z}} = μ$ . Consequently its Beltrami coefficient is $μ_{f} \equiv μ$ and its analytic maximal dilatation is $K_{f} = \frac{1 + ∣ μ ∣}{1 - ∣ μ ∣}$ (The Beltrami coefficient and the maximal dilatation, The Wirtinger derivatives $\partial_{z} f$ and $\partial_{\overset{ˉ}{z}} f$ , and antiholomorphic functions).

(b) The unit circle maps to an ellipse with semiaxes $1 + ∣ μ ∣$ and $1 - ∣ μ ∣$ . Their ratio is $K_{f}$ ; the map is conformal exactly when $μ = 0$ .

(c) For every round annulus $A (r, R) = {r < ∣ z ∣ < R}$ with $0 \leq r < R \leq \infty$ (Annuli in the complex plane), its image is the ring between homothetic ellipses when $0 < r < R < \infty$ . When $r = 0$ its inner complementary component is the puncture ${0}$ ; when $R = \infty$ its outer complementary component in the sphere is ${\infty}$ . Let $Γ_{r, R}$ join the two annular ends, using the end-path convention of An analytically quasiconformal homeomorphism distorts quadrilateral moduli by at most K (equivalently the boundary-joining family for finite positive radii). Then $\frac{1}{K_{f}} λ (Γ_{r, R}) \leq λ (f Γ_{r, R}) \leq K_{f} λ (Γ_{r, R}), \frac{1}{K_{f}} μ (Γ_{r, R}) \leq μ (f Γ_{r, R}) \leq K_{f} μ (Γ_{r, R})$ by An analytically quasiconformal homeomorphism distorts quadrilateral moduli by at most K, where $μ (Γ) = 1 / λ (Γ)$ denotes the library's curve-family modulus. In particular, for the Beltrami parameter $μ = 1 / 3$ and $A (1, e^{2 π})$ , $K_{f} = 2$ and $λ (Γ_{1, e^{2 π}}) = μ (Γ_{1, e^{2 π}}) = 1$ , so the guaranteed distortion interval is $[1 / 2, 2]$ .

(d) The map and its restriction $f ∣_{Ω} : Ω \to f (Ω)$ for every complex domain $Ω$ are $K_{f}$ -quasiconformal in both the analytic and geometric definitions (The ACL and Sobolev analytic definition of quasiconformality, Orientation-preserving homeomorphisms and the geometric definition of quasiconformality).

Facts & Assumptions

Given: Choice, $μ \in C$ with $κ : = ∣ μ ∣ < 1$ , the displayed real-linear map, and the path-family conventions of Extremal length and the curve-family modulus of a path family.

[F1]

Solving $w = z + μ \overline{z}$ together with $\overline{w} = \overline{z} + \overline{μ} z$ gives the stated inverse because $1 - κ^{2} > 0$ . The real determinant is $1 - κ^{2}$ , so the map is invertible and orientation-preserving (A real linear isomorphism preserves or reverses orientation according to the sign of its determinant, Smooth orientation sign is the local integral homology multiplier).

[F2]

Direct Wirtinger differentiation gives $f_{z} = 1$ and $f_{\overset{ˉ}{z}} = μ$ . Thus $μ_{f} = μ$ and the analytic maximal dilatation is $(1 + κ) / (1 - κ)$ (The Wirtinger derivatives $\partial_{z} f$ and $\partial_{\overset{ˉ}{z}} f$ , and antiholomorphic functions, The Beltrami coefficient and the maximal dilatation).

[F3]

Writing $μ = κ e^{i θ}$ and rotating the output by $e^{- i θ / 2}$ gives $e^{- i θ / 2} f (e^{i t}) = (1 + κ) \cos (t - θ / 2) + i (1 - κ) \sin (t - θ / 2) .$ These are the semiaxes of the image ellipse; their ratio equals the value in [F2].

[F4]

Analytic $K_{f}$ -quasiconformality gives both quadrilateral and annular inequalities for extremal length and its reciprocal modulus with constant $K_{f}$ (An analytically quasiconformal homeomorphism distorts quadrilateral moduli by at most K). For $0 < r < R < \infty$ , $λ (Γ_{r, R}) = (2 π)^{- 1} \log (R / r)$ and $μ (Γ_{r, R}) = 1 / λ (Γ_{r, R})$ (Extremal length of the rectangle and of the round annulus).

[F5]

The image of an open connected set under this invertible linear homeomorphism is open and connected, hence a complex domain; the analytic inequality and the geometric quadrilateral bounds restrict to that image (A complex domain is a nonempty connected open subset of $C$ , The ACL and Sobolev analytic definition of quasiconformality, Orientation-preserving homeomorphisms and the geometric definition of quasiconformality, [F1], [F2], [F4]).

Proof

technique · solve the real-linear inverse system, compute the Wirtinger data and ellipse axes, then apply the exact annular modulus theorem

1.1F1givenalgebra

The equations $w = z + μ \overline{z}$ and $\overline{w} = \overline{z} + \overline{μ} z$ imply $w - μ \overline{w} = (1 - ∣ μ ∣^{2}) z$ , so [F1] gives the displayed inverse. Since $1 - ∣ μ ∣^{2} > 0$ , the real determinant is positive; thus $f$ is an orientation-preserving invertible real-linear map and therefore a homeomorphism of $C$ onto itself.

1.2F2F3given

Write $μ = κ e^{i θ}$ . The parametrization in [F3] identifies the image of the unit circle with an ellipse of semiaxes $1 + κ$ and $1 - κ$ , so their ratio is $(1 + κ) / (1 - κ) = K_{f}$ . Also $f_{\overset{ˉ}{z}} = μ$ ; therefore $f$ is holomorphic exactly when $μ = 0$ , in which case it is the identity and conformal.

2.1F2step 1.1algebra

Differentiating $f (z) = z + μ \overset{ˉ}{z}$ gives $f_{z} = 1$ and $f_{\overset{ˉ}{z}} = μ$ . Since this smooth map belongs to $W_{l o c}^{1, 2}$ , the inequality $∣ f_{\overset{ˉ}{z}} ∣ / ∣ f_{z} ∣ = κ < 1$ makes it analytically $K_{f}$ -quasiconformal, and [F2] yields $μ_{f} \equiv μ$ and $K_{f} = (1 + κ) / (1 - κ)$ .

3.1F1F3F4step 2.1givenalgebra

For $0 < r < R < \infty$ , linearity and [F3] send the two boundary circles to homothetic ellipses; homeomorphism sends the region between them onto the region between those ellipses. If $r = 0$ , the omitted origin remains the origin. The bound $∣ f (z) ∣ \geq (1 - κ) ∣ z ∣$ shows that $f$ extends to infinity with $f (\infty) = \infty$ , so $R = \infty$ gives an ellipse exterior, or the punctured plane when also $r = 0$ . The map transports the two annular ends and their path families, and [F4] gives both distortion bounds in every case with its end-path convention. For $μ = 1 / 3$ , $K_{f} = 2$ ; the finite radii $r = 1$ , $R = e^{2 π}$ give $λ = μ (Γ) = 1$ , so each target quantity lies in $[1 / 2, 2]$ .

4.1F1F2F4F5given∎

By [F5], the restriction to any complex domain remains a homeomorphism onto a complex domain, retains the same constant Wirtinger derivatives and analytic inequality, and satisfies the geometric quadrilateral bounds for every quadrilateral compactly contained in that domain. Hence both definitions hold with constant $K_{f}$ on the plane and on every such restriction.

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Sources

Christopher J. Bishop, Quasiconformal Mappings (Stony Brook Math 627 lecture notes) (standard reference, not scraped)
Mikhail Lyubich, Conformal Geometry and Dynamics of Quadratic Polynomials, vol. I (book draft, Stony Brook) (standard reference, not scraped)