Complex & Holomorphic Function Theory

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Abstractions about the analytic structure of holomorphic and harmonic functions, including domains and extension results for holomorphic functions (domain of holomorphy, Ohsawa-Takegoshi L2 extension theorem, Remmert-Stein theorem), pairing of real and complex structure (harmonic conjugate, zeros and poles), and singularity invariants (Milnor number).

8 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Behnke–Stein Open Riemann Surface Theorem — Every connected noncompact Riemann surface is Stein, with global holomorphic separation and convexity.
• Domain of holomorphy — A domain in several complex variables that supports a holomorphic function unable to extend across any strictly larger connected domain, equivalently a holomorphically convex natural domain.
• Harmonic conjugate — Pair real harmonic functions whose gradients satisfy the Cauchy-Riemann rotation so they form the real and imaginary parts of one holomorphic function, subject to global topological existence and additive-constant ambiguity.
• Hartogs's Extension Theorem — Extend a holomorphic function uniquely across a compact hole in a connected domain of at least two complex dimensions.
• Milnor number — Measure an isolated hypersurface singularity by the finite dimension of its local Jacobian algebra, equivalently the number of middle-dimensional spheres in its Milnor fiber.
• Ohsawa–Takegoshi L2 extension theorem — A theorem extending square-integrable holomorphic functions from a complex submanifold with controlled norm.
• Remmert–Stein Theorem — Under a strict component-dimension gap, the closure across a lower-dimensional analytic exceptional set of an analytic subset remains analytic.
• Zeros and Poles — This induces a duality between zeros and poles, that is fundamental for the study of meromorphic functions.