Domains of holomorphy and holomorphic convexity

Intuition Beginner

In one complex variable you can carve almost any region you like out of the plane, and there will always be a holomorphic function that lives exactly on that region and refuses to extend past its boundary. Pick a single boundary point and the function $1/(z-p)$ already blows up there. So in one variable the shape of a region puts no restriction on the functions it can carry.

Several variables behave differently, and the difference is the whole subject. In two or more complex dimensions, some regions are "leaky": every function holomorphic on them is secretly already defined on a larger region. You cannot stop the functions at the intended wall; they spill outward on their own.

A domain of holomorphy is a region with no leak. It is exactly tight: there is at least one holomorphic function on it that genuinely cannot be continued past any part of the boundary. These are the natural homes for complex function theory in higher dimensions.

Visual Beginner

Picture a solid ball in three-dimensional space with a small marble removed from its centre. In the world of several complex variables, any function defined on the ball-minus-marble automatically fills in the marble: the hole heals itself. The ball-with-hole is leaky, so it is not a domain of holomorphy. The full ball, with nothing removed, has no leak and is a genuine domain of holomorphy.

The picture below shows a region together with its "holomorphic convex hull": the extra shaded core is the set of points that every function is forced to control once it controls the outer shell. When that core stays safely inside the region, the region is tight; when the core pushes out to the wall, the region leaks.

Worked example Beginner

Take two-dimensional complex space and remove the single point at the origin. Call this region $X$. Is $X$ a domain of holomorphy?

The Hartogs phenomenon answers no. Any function holomorphic on the punctured space extends holomorphically across the missing origin. You can see the mechanism with a small picture: surround the origin by a thin spherical shell sitting inside $X$. A function holomorphic on the shell can be averaged over the shell to produce a value at the centre that matches up smoothly. The hole at the origin gets filled for free.

So $X$ is leaky. Every holomorphic function on it already lives on the larger region with the origin put back. The origin is not a real boundary for function theory; it is a hole that heals.

Contrast this with the full two-dimensional ball, with nothing removed. There is no hole to heal, and one can build a function on the ball that genuinely stops at the boundary sphere. The full ball is a domain of holomorphy; the punctured space is not.

What this tells us: in several variables, the geometry of a region decides whether function theory on it is "honest". Holes and dents can force functions to extend, and a region is a domain of holomorphy exactly when no such forced extension exists.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $\Omega \subseteq {\mathbb{C}}^n$ is a nonempty open set and $\mathcal{O}(\Omega )$ is the ring of holomorphic functions on $\Omega$ in the sense of 06.07.01. Write $K⋐\Omega$ to mean $K$ is relatively compact in $\Omega$: the closure $\overline{K}$ is compact and contained in $\Omega$. Let $d(z,\partial \Omega )$ denote the Euclidean distance from $z$ to the boundary of $\Omega$ (with the convention $d(z,\partial \Omega )=+\infty$ if $\Omega ={\mathbb{C}}^n$).

Definition (domain of holomorphy). A domain $\Omega \subseteq {\mathbb{C}}^n$ is a domain of holomorphy if there do not exist nonempty open sets ${\Omega }_1,{\Omega }_2\subseteq {\mathbb{C}}^n$ with ${\Omega }_1\subseteq {\Omega }_2\cap \Omega$, ${\Omega }_2$ connected, ${\Omega }_2⊈\Omega$, such that for every $f\in \mathcal{O}(\Omega )$ there is $\tilde{f}\in \mathcal{O}({\Omega }_2)$ with $\tilde{f}=f$ on ${\Omega }_1$. Informally: no single larger connected set receives a simultaneous holomorphic extension of all of $\mathcal{O}(\Omega )$.

Definition (holomorphically convex hull). For $K\subseteq \Omega$ compact, the holomorphically convex hull of $K$ in $\Omega$ is

$${\hat{K}}_{\Omega }=\{z\in \Omega :|f(z)|\le \underset{w\in K}{\sup }|f(w)|\text{ for all }f\in \mathcal{O}(\Omega )\}.$$

The hull is closed in $\Omega$ and bounded (testing against the coordinate functions $z_j$ and $-z_j$ shows ${\hat{K}}_{\Omega }$ lies in the closed convex hull of $K$, hence is bounded). It always contains $K$.

Definition (holomorphically convex domain). $\Omega$ is holomorphically convex if ${\hat{K}}_{\Omega }⋐\Omega$ for every compact $K⋐\Omega$ — equivalently, $K\mapsto {\hat{K}}_{\Omega }$ sends compact sets to relatively compact sets.

Two reformulations of holomorphic convexity will be used. First, $\Omega$ is holomorphically convex if and only if for every compact $K⋐\Omega$ one has $d({\hat{K}}_{\Omega },\partial \Omega )=d(K,\partial \Omega )>0$. Second, $\Omega$ is holomorphically convex if and only if there is a holomorphic exhaustion: an increasing sequence of compacts $K_1⋐K_2⋐\cdots$ with $\bigcup K_j=\Omega$ and each $K_j$ holomorphically convex, i.e. ${\widehat{(K_j)}}_{\Omega }=K_j$.

Counterexamples to common slips

• The hull ${\hat{K}}_{\Omega }$ depends on $\Omega$, not on $K$ alone. Enlarging $\Omega$ enlarges the test family $\mathcal{O}(\Omega )$, which can only shrink the hull. Writing $\hat{K}$ without naming the ambient domain is a category error.

• Holomorphic convexity is strictly weaker than ordinary (linear) convexity. The hull is taken against all holomorphic functions, not just the real-linear ones; an annular shell in $\mathbb{C}$ is holomorphically convex while failing to be convex.

• Boundedness of ${\hat{K}}_{\Omega }$ is automatic; relative compactness inside $\Omega$ is the real condition. The hull can be bounded yet press against $\partial \Omega$, and that is exactly the leak that disqualifies $\Omega$.

Key theorem with proof Intermediate+

Theorem (Cartan–Thullen 1932). For a domain $\Omega \subseteq {\mathbb{C}}^n$ the following are equivalent:

1. $\Omega$ is a domain of holomorphy;
2. $\Omega$ is holomorphically convex, i.e. ${\hat{K}}_{\Omega }⋐\Omega$ for every compact $K⋐\Omega$;
3. $\Omega$ is a domain of existence: there is a single $f\in \mathcal{O}(\Omega )$ that does not extend holomorphically across any boundary point of $\Omega$.

The proof rests on one estimate, which we isolate first ^{[Krantz Ch. 2]}.

Lemma (Cauchy radius estimate on the hull). Let $K⋐\Omega$ be compact and set $\delta =d(K,\partial \Omega )$. For every $z\in {\hat{K}}_{\Omega }$ and every multi-index $\alpha$,

$$\frac{|{\partial }^{\alpha }f(z)|}{\alpha !}{r}^{|\alpha |}\le \underset{w\in K_{\delta }}{\sup }|f(w)|,f\in \mathcal{O}(\Omega ),$$

for every polyradius with all entries $<\delta$, where $r^{|\alpha |}$ abbreviates the monomial $r_1^{{\alpha }_1}\cdots r_n^{{\alpha }_n}$ and $K_{\delta }=\{w:d(w,K)\le {\delta }^{\prime }\}$ for any ${\delta }^{\prime }<\delta$. Consequently the Taylor series of any $f\in \mathcal{O}(\Omega )$ centred at $z\in {\hat{K}}_{\Omega }$ converges on the polydisc of polyradius $(\delta ,\dots ,\delta )$, so $f$ extends holomorphically to ${\bigcup }_{z\in {\hat{K}}_{\Omega }}P(z,\delta )$, where $P(z,\delta )$ is the open polydisc of polyradius $(\delta ,\dots ,\delta )$ about $z$.

Proof of Lemma. Fix a polyradius $\rho =({\rho }_1,\dots ,{\rho }_n)$ with each ${\rho }_j<\delta$ and pick ${\delta }^{\prime }$ with ${\max }_j{\rho }_j<{\delta }^{\prime }<\delta$, so that the closed polydisc $\overline{P(w,\rho )}\subseteq \Omega$ for every $w\in K$. The several-variable Cauchy estimate on the polydisc 06.07.01 gives, for each $w\in K$,

$$\frac{|{\partial }^{\alpha }f(w)|}{\alpha !}\le \frac{{\sup }_{{\partial }_{0}P(w,\rho )}|f|}{{\rho }^{\alpha }}\le \frac{M}{{\rho }^{\alpha }},M:=\underset{w^{\prime }\in K_{\delta }}{\sup }|f(w^{\prime })|,$$

where ${\partial }_0$ is the distinguished boundary (the product of the bounding circles). For each fixed multi-index $\alpha$, the function $g_{\alpha }:={\partial }^{\alpha }f/\alpha !$ is itself holomorphic on $\Omega$, so the inequality $|g_{\alpha }|\le M/{\rho }^{\alpha }$ on $K$ propagates to the hull: by definition of ${\hat{K}}_{\Omega }$, $|g_{\alpha }(z)|\le {\sup }_K|g_{\alpha }|\le M/{\rho }^{\alpha }$ for every $z\in {\hat{K}}_{\Omega }$. Rearranging gives the stated bound with $r=\rho$. Letting the ${\rho }_j\uparrow \delta$ shows the Taylor series of $f$ about any $z\in {\hat{K}}_{\Omega }$ converges on $P(z,\delta )$, and the polydisc sum is a holomorphic extension. $\square$

Proof of Theorem. We prove (2) $\Rightarrow$ (3) $\Rightarrow$ (1) $\Rightarrow$ (2).

(2) $\Rightarrow$ (3). Assume $\Omega$ is holomorphically convex. Choose a holomorphic exhaustion $K_1⋐K_2⋐\cdots$ with ${\widehat{(K_j)}}_{\Omega }=K_j$ and $\bigcup K_j=\Omega$. Pick a countable set $\{z_m\}\subseteq \Omega$ that is dense in $\partial \Omega$ in the sense that every boundary point is a limit of a subsequence, and such that each $z_m$ lies in $K_{j(m)+1}\setminus K_{j(m)}$ with $j(m)\to \infty$; one arranges this by selecting, in each shell $K_{j+1}\setminus K_j$, finitely many points approaching the part of $\partial \Omega$ within distance $1/j$. Because ${\widehat{(K_j)}}_{\Omega }=K_j$ and $z_m\notin K_{j(m)}$, there is $f_m\in \mathcal{O}(\Omega )$ with ${\sup }_{K_{j(m)}}|f_m|<1$ but $f_m(z_m)>2^m\cdot (1+{\sup }_{K_{j(m)}}|f_m|)$; normalising, assume ${\sup }_{K_{j(m)}}|f_m|\le 2^{-m}$ and $|f_m(z_m)|$ exceeds $m+{\sum }_{k<m}|f_k(z_m)|$ after multiplying by a large power. Form

$$f=\prod _{m=1}^{\infty }(1-\frac{f_m}{f_m(z_m)}{)}^m.$$

The product converges locally uniformly on $\Omega$ since on each $K_j$ the tail factors differ from $1$ by at most $2^{-m}$ in modulus, so $f\in \mathcal{O}(\Omega )$. By construction $f$ vanishes to order $\ge m$ at $z_m$. If $f$ extended holomorphically to a connected neighbourhood of some $p\in \partial \Omega$, then a subsequence $z_{m_k}\to p$ would force the extension to vanish to unbounded order at $p$, hence vanish identically near $p$ and therefore on the component of $\Omega$ — impossible since $f\not\equiv 0$. Thus $f$ has $\Omega$ as its exact domain of existence, giving (3).

(3) $\Rightarrow$ (1). A domain of existence is a domain of holomorphy: the single function $f$ of (3) already obstructs any simultaneous extension, since extending all of $\mathcal{O}(\Omega )$ to a larger connected ${\Omega }_2⊈\Omega$ would in particular extend $f$ across a boundary point, contradicting (3).

(1) $\Rightarrow$ (2). We prove the contrapositive. Suppose $\Omega$ is not holomorphically convex: there is a compact $K⋐\Omega$ with $\delta :=d(K,\partial \Omega )>0$ yet a point $z_0\in {\hat{K}}_{\Omega }$ with $d(z_0,\partial \Omega )<\delta$. By the Lemma, every $f\in \mathcal{O}(\Omega )$ extends holomorphically to the polydisc $P(z_0,\delta )$, whose radius exceeds $d(z_0,\partial \Omega )$, so $P(z_0,\delta )⊈\Omega$. Taking ${\Omega }_1$ to be a small polydisc about $z_0$ inside $\Omega$ and ${\Omega }_2=P(z_0,\delta )$ exhibits a simultaneous extension of all of $\mathcal{O}(\Omega )$ to a connected set not contained in $\Omega$. Hence $\Omega$ is not a domain of holomorphy. Contrapositively, (1) implies (2). $\square$

Bridge. The Cartan–Thullen equivalence builds toward the analytic theory that fills the rest of this chapter, and the Cauchy radius estimate appears again in the $L^2$ existence theorem for the $\bar{\partial }$ operator, where the same distance-to-the-boundary controls the weight. The foundational reason the three conditions coincide is that holomorphic convexity is exactly the obstruction to the polydisc-extension built from the Cauchy estimate: this is exactly the statement that the hull cannot outrun the boundary by even one Taylor radius. The equivalence generalises: replacing the test family $\mathcal{O}(\Omega )$ by the plurisubharmonic functions of 06.10.02 (when it ships) converts holomorphic convexity into pseudoconvexity, and the central insight is that the bridge is the distance function $-\log d(z,\partial \Omega )$, whose plurisubharmonicity is the analytic certificate that a domain is a domain of holomorphy. Putting these together, Cartan–Thullen identifies the function-theoretic notion (no leak) with the geometric-convexity notion (hull stays inside), and the solution of the Levi problem will identify both with pseudoconvexity.

Exercises Intermediate+

Advanced results Master

The Cartan–Thullen theorem is the first of two great equivalences in the structure theory of ${\mathbb{C}}^n$. The second, the solution of the Levi problem, will identify both conditions with pseudoconvexity through the $\bar{\partial }$ machinery. The results below sharpen the convexity picture and connect it to the analytic tools that follow.

Holomorphic convexity via the distance function. For $\Omega \subseteq {\mathbb{C}}^n$ open, define $\phi (z)=-\log d(z,\partial \Omega )$. Cartan–Thullen has a quantitative refinement: $\Omega$ is a domain of holomorphy if and only if for every compact $K⋐\Omega$, $d({\hat{K}}_{\Omega },\partial \Omega )=d(K,\partial \Omega )$. The Lemma of the Key-theorem section is precisely the inequality $d({\hat{K}}_{\Omega },\partial \Omega )\ge d(K,\partial \Omega )$, which holds on every domain; the reverse inequality $\le$ is automatic since $K\subseteq {\hat{K}}_{\Omega }$. The content of Cartan–Thullen is that this radius equality is equivalent to holomorphic convexity, because the failure of relative compactness of ${\hat{K}}_{\Omega }$ is the same as a point of the hull with strictly smaller boundary distance — the leak. This reformulation is the entry point to the plurisubharmonic theory of 06.10.02: $\phi$ is plurisubharmonic exactly when $\Omega$ is pseudoconvex, and the equivalence of pseudoconvexity with holomorphic convexity is the Levi problem.

Convexity with respect to a family. The hull construction is functorial in the test family. For any subset $\mathcal{F}\subseteq \mathcal{O}(\Omega )$ closed under products, define ${\hat{K}}_{\mathcal{F}}=\{z:|f(z)|\le {\sup }_K|f|,f\in \mathcal{F}\}$. Taking $\mathcal{F}$ to be the polynomials yields polynomial convexity; taking $\mathcal{F}=\mathcal{O}({\mathbb{C}}^n)$ yields holomorphic (Runge) convexity; taking $\mathcal{F}=\mathcal{O}(\Omega )$ yields the hull above. A compact $K\subseteq {\mathbb{C}}^n$ is polynomially convex if ${\hat{K}}_{\text{poly}}=K$; the Oka–Weil theorem (every function holomorphic near a polynomially convex compact is a uniform limit of polynomials) is the approximation analogue of Cartan–Thullen and feeds the several-variable Runge theory.

Stability properties. Domains of holomorphy are closed under several operations, each provable directly from holomorphic convexity. Arbitrary intersections (of connected components) of domains of holomorphy are domains of holomorphy; increasing unions $\Omega ={\bigcup }_j{\Omega }_j$ with each ${\Omega }_j$ a domain of holomorphy and ${\Omega }_j⋐{\Omega }_{j+1}$ are domains of holomorphy (Behnke–Stein, the dimension-one case being 06.09.03); products ${\Omega }_1\times {\Omega }_2$ are domains of holomorphy; and the biholomorphic image of a domain of holomorphy is a domain of holomorphy, since the hull is biholomorphically invariant.

The role in the $\bar{\partial }$ programme. The deepest reason holomorphic convexity matters is cohomological: $\Omega$ is a domain of holomorphy if and only if $H^q(\Omega ,\mathcal{O})=0$ for all $q\ge 1$ (Cartan's Theorem B for the Stein domains $\Omega$ defines). The vanishing is proved analytically by solving $\bar{\partial }u=f$ with $L^2$ estimates on pseudoconvex domains — the Hörmander theory — and the Cauchy radius estimate is the first instance of the distance-controlled estimates that recur there. The equivalence of "domain of holomorphy", "holomorphically convex", and "Stein" makes Cartan–Thullen the gateway from elementary ${\mathbb{C}}^n$ function theory to the full Oka–Cartan and Hörmander machinery.

Synthesis. Cartan–Thullen is the foundational reason several-complex-variable function theory has a clean notion of natural domain at all: the geometric condition that the holomorphically convex hull never outruns the boundary is exactly the function-theoretic condition that no simultaneous extension exists, and this is exactly the radius equality $d({\hat{K}}_{\Omega },\partial \Omega )=d(K,\partial \Omega )$ delivered by the Cauchy estimate. The single estimate generalises in two directions that organise the rest of the subject: replacing the test family with plurisubharmonic functions identifies holomorphic convexity with pseudoconvexity, and the distance function $-\log d(z,\partial \Omega )$ is dual to the hull in the sense that its plurisubharmonicity is the infinitesimal certificate of the global no-leak property. Putting these together, the central insight is that the cohomological vanishing $H^{q\ge 1}(\Omega ,\mathcal{O})=0$, the holomorphic convexity of $\Omega$, and the existence of a single non-extendable function are three faces of one structure; the Levi problem completes the circle by adding pseudoconvexity, and the Hörmander $L^2$ theory of 06.10.04 (when it ships) is the analytic engine that proves the hardest implication. The bridge is, throughout, the boundary distance — measured against the polydisc radius for Cartan–Thullen, against the weight $e^{-\phi }$ for Hörmander.

Full proof set Master

Proposition 1 (the hull is compact in a domain of holomorphy). If $\Omega$ is a domain of holomorphy and $K⋐\Omega$ is compact, then ${\hat{K}}_{\Omega }$ is compact and ${\widehat{({\hat{K}}_{\Omega })}}_{\Omega }={\hat{K}}_{\Omega }$.

Proof. By Cartan–Thullen, $\Omega$ holomorphically convex gives ${\hat{K}}_{\Omega }⋐\Omega$; the hull is closed in $\Omega$ (an intersection of closed sets $\{|f|\le {\sup }_K|f|\}$) and relatively compact, hence its closure $\overline{{\hat{K}}_{\Omega }}\subseteq \Omega$ is compact, and being closed in $\Omega$ it equals ${\hat{K}}_{\Omega }$. For idempotence, the inclusion ${\hat{K}}_{\Omega }\subseteq {\widehat{({\hat{K}}_{\Omega })}}_{\Omega }$ is general. Conversely, for $f\in \mathcal{O}(\Omega )$, ${\sup }_{{\hat{K}}_{\Omega }}|f|={\sup }_K|f|$ because each $f$ already satisfies $|f|\le {\sup }_K|f|$ on ${\hat{K}}_{\Omega }$ by definition; thus any $z$ with $|f(z)|\le {\sup }_{{\hat{K}}_{\Omega }}|f|$ for all $f$ also satisfies $|f(z)|\le {\sup }_K|f|$, giving ${\widehat{({\hat{K}}_{\Omega })}}_{\Omega }\subseteq {\hat{K}}_{\Omega }$. $\square$

Proposition 2 (radius equality on a domain of holomorphy). If $\Omega$ is a domain of holomorphy, then $d({\hat{K}}_{\Omega },\partial \Omega )=d(K,\partial \Omega )$ for every compact $K⋐\Omega$.

Proof. The inequality $d({\hat{K}}_{\Omega },\partial \Omega )\le d(K,\partial \Omega )$ holds since $K\subseteq {\hat{K}}_{\Omega }$. For the reverse, the Cauchy radius estimate of the Key-theorem Lemma shows every $f\in \mathcal{O}(\Omega )$ extends to the polydisc $P(z,\delta )$ for each $z\in {\hat{K}}_{\Omega }$, with $\delta =d(K,\partial \Omega )$. If some $z_0\in {\hat{K}}_{\Omega }$ had $d(z_0,\partial \Omega )<\delta$, the common polydisc extension would carry all of $\mathcal{O}(\Omega )$ to a connected set meeting the complement of $\Omega$, contradicting that $\Omega$ is a domain of holomorphy. Hence $d(z,\partial \Omega )\ge \delta$ for all $z\in {\hat{K}}_{\Omega }$, giving $d({\hat{K}}_{\Omega },\partial \Omega )\ge d(K,\partial \Omega )$. $\square$

Proposition 3 (products). If ${\Omega }_1\subseteq {\mathbb{C}}^{n_1}$ and ${\Omega }_2\subseteq {\mathbb{C}}^{n_2}$ are domains of holomorphy, so is ${\Omega }_1\times {\Omega }_2\subseteq {\mathbb{C}}^{n_1+n_2}$.

Proof. Let $K⋐{\Omega }_1\times {\Omega }_2$ be compact and let ${\pi }_i$ be the coordinate projections, so $K\subseteq {\pi }_1(K)\times {\pi }_2(K)$ with each ${\pi }_i(K)⋐{\Omega }_i$ compact. Functions of the form $(z,w)\mapsto f(z)g(w)$ with $f\in \mathcal{O}({\Omega }_1)$, $g\in \mathcal{O}({\Omega }_2)$ lie in $\mathcal{O}({\Omega }_1\times {\Omega }_2)$; testing the hull against $f(z)\cdot 1$ and $1\cdot g(w)$ gives ${\hat{K}}_{{\Omega }_1\times {\Omega }_2}\subseteq {\widehat{({\pi }_{1}K)}}_{{\Omega }_1}\times {\widehat{({\pi }_{2}K)}}_{{\Omega }_2}$. By hypothesis each factor hull is relatively compact in its ${\Omega }_i$, so the product is relatively compact in ${\Omega }_1\times {\Omega }_2$. Holomorphic convexity follows, and Cartan–Thullen gives the result. $\square$

Proposition 4 (Behnke–Stein increasing unions). If ${\Omega }_1⋐{\Omega }_2⋐\cdots$ are domains of holomorphy with ${\Omega }_j⋐{\Omega }_{j+1}$ and the pairs Runge (every $f\in \mathcal{O}({\Omega }_j)$ is a locally uniform limit of restrictions from $\mathcal{O}({\Omega }_{j+1})$), then $\Omega ={\bigcup }_j{\Omega }_j$ is a domain of holomorphy.

Proof. Let $K⋐\Omega$; then $K\subseteq {\Omega }_{j_0}$ for some $j_0$ by compactness. The Runge condition forces ${\hat{K}}_{\Omega }\cap {\Omega }_j={\hat{K}}_{{\Omega }_j}$ for $j\ge j_0$: a Runge pair has ${\hat{K}}_{{\Omega }_{j+1}}\cap {\Omega }_j={\hat{K}}_{{\Omega }_j}$, and the test family stabilises in the limit. Each ${\hat{K}}_{{\Omega }_j}$ is relatively compact in ${\Omega }_j⋐\Omega$, and the radius equality of Proposition 2 applied in ${\Omega }_{j_0}$ gives a uniform lower bound $d({\hat{K}}_{\Omega },\partial \Omega )\ge d(K,\partial {\Omega }_{j_0})>0$ that does not degrade as $j\to \infty$. Hence ${\hat{K}}_{\Omega }⋐\Omega$, and $\Omega$ is holomorphically convex, so a domain of holomorphy. The dimension-one instance is 06.09.03. $\square$

Connections Master

• Holomorphic functions of several variables 06.07.01. This unit supplies the polydisc Cauchy integral, multi-index calculus, and the Cauchy estimates on the distinguished boundary that drive the radius Lemma at the heart of Cartan–Thullen. Domains of holomorphy are the natural domains of definition for the function theory built there.

• Hartogs phenomenon 06.07.02. The Hartogs extension is the prototype leak: it is what makes ${\mathbb{C}}^n\setminus \{0\}$ and the Hartogs figure fail to be domains of holomorphy, and it is the motivating obstruction whose absence Cartan–Thullen characterises. Holomorphic convexity is the precise condition ruling out every Hartogs-type extension at once.

• Analytic continuation 06.01.04. A domain of holomorphy is defined by the impossibility of simultaneous analytic continuation of the whole ring $\mathcal{O}(\Omega )$; the several-variable theory is exactly the place where the one-variable freedom of continuation across boundary points breaks down.

• Behnke–Stein theorem 06.09.03. The increasing-union stability of domains of holomorphy (Proposition 4) is the higher-dimensional analogue of the dimension-one Behnke–Stein theorem on Riemann surfaces; both express that exhaustion by Runge subdomains preserves the Stein property.

• Cousin I (additive) 06.09.04. The interpolation and pole-prescription problems solvable on domains of holomorphy rest on the cohomological vanishing $H^1(\Omega ,\mathcal{O})=0$ that holomorphic convexity guarantees; Cousin I in ${\mathbb{C}}^n$ is the several-variable lift of this dimension-one unit.

• $\bar{\partial }$ as a Hilbert-space PDE 06.04.05. The dimension-one $\bar{\partial }$ existence theory is the seed of the Hörmander $L^2$ method on pseudoconvex domains; the boundary distance controlling the Cauchy radius estimate here is the same quantity controlling the weight $e^{-\phi }$ there, and the cohomological characterisation of domains of holomorphy is proved by that method.

Historical & philosophical context Master

The decisive theorem was proved by Henri Cartan and Peter Thullen in 1932 in Zur Theorie der Singularitäten der Funktionen mehrerer komplexen Veränderlichen ^{[Cartan–Thullen 1932]} (Mathematische Annalen 106, 617–647). Their paper isolated the holomorphically convex hull ${\hat{K}}_{\Omega }$ and proved that relative compactness of the hull is equivalent to the domain being a domain of holomorphy, supplying the Cauchy-estimate argument that converts a point of the hull near the boundary into a forced extension. The result gave the first intrinsic, function-theoretic characterisation of the domains that are natural for several-variable theory, replacing the case-by-case constructions of Hartogs and Friedrich Hartogs's school.

The phenomenon the theorem explains was discovered by Hartogs in 1906 ^{[Hartogs 1906]} (Mathematische Annalen 62, 1–88), who observed that holomorphic functions on a spherical shell in ${\mathbb{C}}^2$ extend across the enclosed ball — the failure of isolated singularities to exist for $n\ge 2$. The contrast with the one-variable theory, where every domain is a domain of holomorphy, made clear that the geometry of $\Omega \subseteq {\mathbb{C}}^n$ constrains its function theory in an essential way, a fact with no dimension-one precedent.

Cartan–Thullen left open the converse question that became the Levi problem, after Eugenio Elia Levi's 1910 study of the boundary condition (the Levi form) that a domain of holomorphy must satisfy: is every pseudoconvex domain a domain of holomorphy? The affirmative answer came in stages — Kiyoshi Oka for $n=2$ in 1942 and the general bounded case in 1953, with Hans-Joachim Bremermann and François Norguet independently completing the unbounded and general cases in 1954 — and was later given a clean analytic proof through Lars Hörmander's 1965 $L^2$ estimates for the $\bar{\partial }$ operator. The holomorphically convex hull of Cartan–Thullen thereby sits at the head of the entire Oka–Cartan and Hörmander development of several-variable complex analysis.

Bibliography Master

@article{CartanThullen1932,
  author  = {Cartan, Henri and Thullen, Peter},
  title   = {Zur {Theorie} der {Singularit\"aten} der {Funktionen}
             mehrerer komplexen {Ver\"anderlichen}: {Regularit\"ats-}
             und {Konvergenzbereiche}},
  journal = {Math. Ann.},
  volume  = {106},
  year    = {1932},
  pages   = {617--647}
}

@article{Hartogs1906,
  author  = {Hartogs, Friedrich},
  title   = {Zur {Theorie} der analytischen {Funktionen} mehrerer
             unabh\"angiger {Ver\"anderlicher}},
  journal = {Math. Ann.},
  volume  = {62},
  year    = {1906},
  pages   = {1--88}
}

@book{KrantzSCV,
  author    = {Krantz, Steven G.},
  title     = {Function Theory of Several Complex Variables},
  edition   = {2},
  series    = {AMS Chelsea Publishing},
  volume    = {340},
  publisher = {American Mathematical Society},
  year      = {2001}
}

@book{HormanderSCV,
  author    = {H\"ormander, Lars},
  title     = {An Introduction to Complex Analysis in Several Variables},
  edition   = {3},
  series    = {North-Holland Mathematical Library},
  volume    = {7},
  publisher = {North-Holland},
  year      = {1990}
}

@book{RangeSCV,
  author    = {Range, R. Michael},
  title     = {Holomorphic Functions and Integral Representations in
               Several Complex Variables},
  series    = {Graduate Texts in Mathematics},
  volume    = {108},
  publisher = {Springer},
  year      = {1986}
}

@article{Oka1953,
  author  = {Oka, Kiyoshi},
  title   = {Sur les fonctions analytiques de plusieurs variables. IX.
             {Domaines} finis sans point critique int\'erieur},
  journal = {Japan. J. Math.},
  volume  = {23},
  year    = {1953},
  pages   = {97--155}
}

@article{Hormander1965,
  author  = {H\"ormander, Lars},
  title   = {{$L^2$} estimates and existence theorems for the
             {$\bar\partial$} operator},
  journal = {Acta Math.},
  volume  = {113},
  year    = {1965},
  pages   = {89--152}
}