LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-08

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A local Jacobian and energy bound for quasiconformal homeomorphisms

Statement

Assume the Axiom of Choice. Let $Ω, Ω^{'} \subseteq C$ be complex domains and let $f : Ω \to Ω^{'}$ be an orientation-preserving $K$ -geometrically quasiconformal homeomorphism, where $K \geq 1$ . Put $k = (K - 1) / (K + 1)$ . Write $\partial_{z} f, \partial_{\overset{ˉ}{z}} f$ for its weak Wirtinger derivatives, $J_{f} = ∣ \partial_{z} f ∣^{2} - ∣ \partial_{\overset{ˉ}{z}} f ∣^{2}$ for its Jacobian, and $∣ D f ∣_{H S}$ for the Hilbert--Schmidt norm of its real weak derivative matrix. Then for every relatively compact Borel set $E \subset Ω$ , $\int_{E} J_{f} d A \leq λ_{2} (f (E)), \int_{E} ∣ D f ∣_{H S}^{2} d A \leq \frac{2 (1 + k^{2})}{1 - k^{2}} λ_{2} (f (E)) .$

Facts & Assumptions

Given: AC, the geometric K-QC homeomorphism and the relatively compact Borel set E.

[F1]

Geometric and analytic K-quasiconformality agree, so the weak Wirtinger derivatives exist and obey $∣ f_{\overset{ˉ}{z}} ∣ \leq k ∣ f_{z} ∣$ (The geometric and analytic definitions of quasiconformality agree, The ACL and Sobolev analytic definition of quasiconformality).

[F2]

The earlier full distortion wrapper proves $∣ f (E) ∣ = \int_{E} J_{f}$ without assuming the present lemma or MRMT (An analytically quasiconformal homeomorphism distorts quadrilateral moduli by at most K, area clause). Its lower inequality suffices here.

[F3]

Expanding the Wirtinger identities gives $∣ D f ∣_{H S}^{2} = 2 (∣ f_{z} ∣^{2} + ∣ f_{\overset{ˉ}{z}} ∣^{2})$ and $J_{f} = ∣ f_{z} ∣^{2} - ∣ f_{\overset{ˉ}{z}} ∣^{2}$ (The Wirtinger derivatives $\partial_{z} f$ and $\partial_{\overset{ˉ}{z}} f$ , and antiholomorphic functions).

Proof

technique · use the earlier proved planar area formula and the Beltrami energy algebra

1.1F1F2given

Apply [F1] to regard f as an analytic K-QC map. The exact Borel-set area formula in [F2] gives $\int_{E} J_{f} = ∣ f (E) ∣$ , hence the claimed lower inequality. No unrecovered Gehring–Lehto source is used; the complete differentiability and signed-degree arguments are in the earlier12 suppliers.

2.1F1F3step 1.1algebra∎

The Beltrami bound gives $J_{f} \geq (1 - k^{2}) ∣ f_{z} ∣^{2}$ , and [F3] gives $∣ D f ∣_{H S}^{2} \leq 2 (1 + k^{2}) ∣ f_{z} ∣^{2} \leq 2 (1 + k^{2}) (1 - k^{2})^{- 1} J_{f}$ . Integrate and use step 1.1 to obtain the stated constant.

Depends on

Used by

Compact subsets of lines and round circles are removable for quasiconformal maps Lemma

Dependency tree · two levels

46 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Mikhail Lyubich, Conformal Geometry and Dynamics of Quadratic Polynomials, vol. I (standard reference, not scraped)
Christopher J. Bishop, Quasiconformal Mappings (Stony Brook Math 627 course notes) (standard reference, not scraped)
F. W. Gehring and O. Lehto, On the total differentiability of functions of a complex variable (standard reference, not scraped)