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An injective holomorphic map has no critical point and is biholomorphic onto its image
Statement
An injective holomorphic map on a complex domain has nowhere-zero derivative and is biholomorphic onto its open image.
Precisely, if is holomorphic and injective on a complex domain, then for every , the set is a complex domain, and is biholomorphic (Biholomorphic maps between complex domains).
Facts & Assumptions
Given: A holomorphic injective map on a complex domain. Injectivity and bijectivity have their set-theoretic meanings (Injection, surjection, bijection).
If is nonconstant and holomorphic on a complex domain and , then , , local injectivity at , and biholomorphy between neighbourhoods of and are equivalent (Holomorphic inverse function theorem and local-degree criterion).
Every nonconstant holomorphic function on a complex domain is an open map (Open mapping theorem for holomorphic functions).
A complex differentiable function is continuous at every point of complex differentiability (Complex differentiability at a point implies continuity there).
A continuous image of a connected space is connected (A continuous image of a connected space is connected, and connectedness is a topological property, claim 1).
Proof
The map is nonconstant because an open complex domain has distinct points and is injective. It is locally injective at every , so [L1] gives and a holomorphic local inverse near .
By [L2], the image is open. By [L3], is continuous, so [L4] makes its image connected; it is therefore a complex domain.
The global set-theoretic inverse agrees near every image point with the holomorphic local inverse from step 1.1. Hence is holomorphic throughout the image.
The map is bijective onto its image, and step 2.1 makes its inverse holomorphic; therefore it is biholomorphic onto the open image.
Depends on
- Holomorphic inverse function theorem and local-degree criterion
- Open mapping theorem for holomorphic functions
- Complex differentiability at a point implies continuity there
- A continuous image of a connected space is connected, and connectedness is a topological property
- Biholomorphic maps between complex domains
- Injection, surjection, bijection
Used by
- Every Riemann surface is a quotient of a simply connected model Corollary
- Meromorphic differentials, orders and residues Definition
- Rectangular lattices, real mapping, and inverse elliptic integrals Example
- The modular lambda function: Y(2) biholomorphic to the twice-punctured plane, and the slit-plane quadrilateral Example
- FALSE: every injective real-differentiable planar map has nonzero Jacobian False statement
- A simply connected Greenian Riemann surface is a disc Lemma
- A simply connected surface without a Green kernel is plane or sphere Lemma
- Finite chartwise triangulation of a compact Riemann surface Lemma
- Local charts and the Riemann surface structure of a modular quotient Lemma
- Conformal covariance of the canonical planar Green kernel Theorem
- Deck transformations preserve the hyperbolic metric Theorem
- Every proper homologically simply connected plane domain is conformally equivalent to the unit disc Theorem
- Green kernel of a simply connected plane domain from a Riemann map Theorem
- The area theorem for exterior univalent functions Theorem
- The j-invariant classifies complex tori Theorem
- The j-invariant uniformizes X(1) Theorem
- Topological classification of compact Riemann surfaces Theorem
Dependency tree · two levels
27 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
- J. Lebl, Guide to Cultivating Complex Analysis, Theorem 5.6.3 (standard reference, not scraped)
- B. V. Shabat, Introduction to Complex Analysis, §1.2 (standard reference, not scraped)