06.10.03 · riemann-surfaces / several-variables

Pseudoconvexity and the Levi form

shipped3 tiersLean: none

Anchor (Master): Levi 1910 / Oka 1942 (originators); Krantz Ch. 2 §2.6; Hörmander §2.6; Range Ch. III

Intuition Beginner

A region in the plane is convex when, for any two of its points, the straight segment between them stays inside. In several complex variables the right notion of "good shape" is weaker and more subtle. You do not test the region against all straight lines. You test it against the special slices that the complex structure cares about: complex lines, the flat two-dimensional sheets that look like a copy of the ordinary plane.

A pseudoconvex region is one that curves the right way along exactly these complex slices, and is allowed to dip along the others. It is the largest class of regions for which several-variable function theory behaves well, and it turns out to be precisely the class with no leak from the previous units: the regions where holomorphic functions genuinely stop at the wall.

The barometer is a single number attached to each boundary point. Slide a complex slice up against the boundary so it touches at one point. If the boundary bends away from the slice, that point is good; if it bends into the slice, the region leaks there. That bending number is the Levi form.

Visual Beginner

Picture the boundary wall of a region in complex space and a flat complex sheet pressed against it, touching at a single point and otherwise staying outside. At a pseudoconvex boundary point the wall curls away from the sheet on both sides, the way the inside of a bowl curls away from a tabletop laid across its rim. At a bad point the wall bulges into the sheet, like a hill poking up through the tabletop, and through that bulge functions can leak out.

The picture below shows the boundary of a region with a complex tangent sheet at one point. The Levi form measures the gap between the wall and the sheet just off the contact point: positive gap means the wall retreats and the point is good; a gap of the wrong sign means the wall advances and the region fails the test there.

The barrier that certifies the whole region at once is the height $- lo g d (z)$ , where $d (z)$ is the distance to the wall. When this barrier curves upward along every complex slice, the region is pseudoconvex everywhere.

Worked example Beginner

Take the unit ball in two complex variables: all points where $∣ z_{1} ∣^{2} + ∣ z_{2} ∣^{2} < 1$ . Is it pseudoconvex?

The wall is the sphere where $∣ z_{1} ∣^{2} + ∣ z_{2} ∣^{2} = 1$ . A natural height function that is negative inside, zero on the wall, and positive outside is $ρ (z) = ∣ z_{1} ∣^{2} + ∣ z_{2} ∣^{2} - 1$ . The bending of the wall along a complex slice is read off from the second-order behaviour of this height. For the ball, that bending number works out to $∣ w_{1} ∣^{2} + ∣ w_{2} ∣^{2}$ for a slice direction $w$ , which is positive for every nonzero direction. The wall retreats from every complex sheet in every direction.

So the ball passes the test at every boundary point with room to spare. It is strictly pseudoconvex: the bending is not merely nonnegative but strictly positive everywhere. This matches the picture of a round bowl, whose inside curls away from any flat sheet laid across it.

Now contrast a region with a flat spot in a complex direction, say a piece of wall that contains a whole complex disc. Along that disc the wall does not retreat at all; the bending number is zero. Such a point is on the boundary of being good. Push the wall slightly the wrong way and the bending turns negative, and at that moment the region leaks.

What this tells us: pseudoconvexity is the demand that the wall never bend into a complex sheet. The ball satisfies it everywhere; a region with a complex-flat wall sits on the edge; and a region whose wall bulges into a complex sheet fails, and through that failure functions escape.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $Ω \subseteq C^{n}$ is a nonempty open set, $δ (z) = d (z, \partial Ω)$ is the Euclidean distance to the boundary, and $PSH (Ω)$ is the cone of plurisubharmonic functions of 06.10.02. We use the sign convention $ρ < 0$ inside $Ω$ throughout, so that the outward direction is where $ρ$ increases.

Definition (Hartogs pseudoconvexity). $Ω$ is (Hartogs) pseudoconvex if the function $- lo g δ (z) = - lo g d (z, \partial Ω)$ is plurisubharmonic on $Ω$ . (When $Ω = C^{n}$ one sets $δ \equiv + \infty$ and declares $Ω$ pseudoconvex.)

This is the global, boundary-distance form. There is an equivalent form in terms of exhaustion functions and, for smooth boundaries, a local form in terms of the Levi form.

Definition (defining function). A $C^{2}$ defining function for a domain $Ω$ with $C^{2}$ boundary near $p \in \partial Ω$ is a real $C^{2}$ function $ρ$ on a neighbourhood $U$ of $p$ with $Ω \cap U = {ρ < 0}$ , $\partial Ω \cap U = {ρ = 0}$ , and $d ρ \neq = 0$ on ${ρ = 0}$ . Writing $\partial ρ = \sum_{j} (\partial ρ / \partial z_{j}) d z_{j}$ , the complex tangent space to $\partial Ω$ at $p$ is

T_{p}^{C} \partial Ω = {w \in C^{n} : j = 1 \sum n \frac{\partial ρ}{\partial z _{j}} (p) w_{j} = 0},

the largest complex-linear subspace of the real tangent hyperplane; it has complex dimension $n - 1$ .

Definition (Levi form and Levi pseudoconvexity). The Levi form of $ρ$ at $p$ is the Hermitian form

L_{ρ} (p; w) = j, k = 1 \sum n \frac{\partial ^{2} ρ}{\partial z _{j} \partial z ˉ _{k}} (p) w_{j} \overset{w}{ˉ}_{k} .

A domain $Ω$ with $C^{2}$ boundary is Levi pseudoconvex at $p \in \partial Ω$ if $L_{ρ} (p; w) \geq 0$ for all $w \in T_{p}^{C} \partial Ω$ , and strictly (or strongly) Levi pseudoconvex at $p$ if $L_{ρ} (p; w) > 0$ for all nonzero $w \in T_{p}^{C} \partial Ω$ . It is Levi pseudoconvex if this holds at every boundary point.

The Levi form on $T_{p}^{C} \partial Ω$ does not depend on the choice of defining function (Proposition 1 of the Full proof set). The value on directions outside $T_{p}^{C} \partial Ω$ does depend on $ρ$ and carries no geometric meaning — this is why the form is read only on the complex tangent space.

Definition (PSH exhaustion). Recall from 06.10.02 that $φ : Ω \to R$ is an exhaustion if every sublevel set $Ω_{c} = {φ < c}$ is relatively compact in $Ω$ , and a PSH exhaustion if additionally $φ \in PSH (Ω)$ .

Counterexamples to common slips

The Levi form is read only on the complex tangent space $T_{p}^{C} \partial Ω$ , never on all of $C^{n}$ . On the full space one can change the sign of $L_{ρ} (p; w)$ for non-tangential $w$ by replacing $ρ$ with $e^{λ ρ} - 1$ ; only the restriction to $T_{p}^{C} \partial Ω$ is invariant.
Pseudoconvexity is strictly weaker than convexity. A bounded convex domain is Levi pseudoconvex, but the converse fails: the worm domains and the Hartogs triangle complements are pseudoconvex yet far from convex. Convexity tests the real Hessian on all real tangent directions; pseudoconvexity tests only the complex Hessian on complex tangent directions.
A boundary may be Levi pseudoconvex with the Levi form vanishing identically on $T_{p}^{C}$ (the weakly pseudoconvex, or Levi-flat, case), as for the boundary ${∣ z_{2} ∣ = 1}$ inside a polydisc. Weak pseudoconvexity is genuinely harder than strict, and many sharp $\overset{ˉ}{\partial}$ estimates degrade there.

Key theorem with proof Intermediate+

Theorem (equivalence of pseudoconvexity notions). Let $Ω \subseteq C^{n}$ be open. Consider the conditions:

$Ω$ is Hartogs pseudoconvex, i.e. $- lo g δ \in PSH (Ω)$ ;
$Ω$ admits a continuous plurisubharmonic exhaustion function;
(if $\partial Ω$ is $C^{2}$ ) $Ω$ is Levi pseudoconvex, i.e. $L_{ρ} (p; w) \geq 0$ for all $p \in \partial Ω$ and all $w \in T_{p}^{C} \partial Ω$ .

Then (1) $\Leftrightarrow$ (2) always, and (1) $\Leftrightarrow$ (3) when $\partial Ω$ is $C^{2}$ .

The proof of (1) $\Leftrightarrow$ (3) rests on relating the Levi form of the defining function $ρ$ to the Levi form of the distance function $δ$ , which differ by a controlled boundary term ^{[Krantz Ch. 2]}.

Lemma (Levi form of a defining function controls $- lo g δ$ near the boundary). Let $Ω$ have $C^{2}$ boundary with defining function $ρ$ normalised so that $∣\nabla ρ ∣ = 1$ on $\partial Ω$ (the signed-distance normalisation, $ρ = - δ$ near $\partial Ω$ up to $C^{2}$ error). For $z$ near $\partial Ω$ and any $w \in C^{n}$ ,

j, k \sum \frac{\partial ^{2} ( - lo g δ )}{\partial z _{j} \partial z ˉ _{k}} (z) w_{j} \overset{w}{ˉ}_{k} = \frac{1}{δ ( z )} L_{ρ} (z; w) + \frac{1}{δ ( z ) ^{2}} j \sum \frac{\partial ρ}{\partial z _{j}} (z) w_{j}^{2} + O (1) ∣ w ∣^{2},

where the $O (1)$ term is bounded uniformly near $\partial Ω$ .

Proof of Lemma. Write $u = - lo g δ = - lo g (- ρ)$ near $\partial Ω$ , valid because $ρ = - δ + O (δ^{2})$ under the normalisation. Differentiating $u = - lo g (- ρ)$ ,

\frac{\partial u}{\partial z _{j}} = \frac{1}{- ρ} \frac{\partial ρ}{\partial z _{j}} = - \frac{1}{ρ} \frac{\partial ρ}{\partial z _{j}}, \frac{\partial ^{2} u}{\partial z _{j} \partial z ˉ _{k}} = - \frac{1}{ρ} \frac{\partial ^{2} ρ}{\partial z _{j} \partial z ˉ _{k}} + \frac{1}{ρ ^{2}} \frac{\partial ρ}{\partial z _{j}} \frac{\partial ρ}{\partial z ˉ _{k}} .

Since $- ρ = δ + O (δ^{2})$ , the first term is $δ^{- 1} L_{ρ} (z; \cdot) + O (1)$ and the second is $δ^{- 2} ∣ \sum_{j} (\partial ρ / \partial z_{j}) w_{j} ∣^{2} + O (δ^{- 1})$ when contracted against $w \otimes \overset{w}{ˉ}$ ; absorbing the lower-order corrections into the $O (1) ∣ w ∣^{2}$ term gives the stated identity. $□$

Proof of Theorem.

(1) $\Rightarrow$ (2). If $Ω = C^{n}$ , take $φ = ∣ z ∣^{2}$ . Otherwise $- lo g δ \in PSH (Ω)$ by hypothesis, and $φ (z) = max (- lo g δ (z), ∣ z ∣^{2})$ is a maximum of two PSH functions, hence PSH (06.10.02), continuous, and an exhaustion: a sublevel set forces both $δ (z) > e^{- c}$ (bounded away from the boundary) and $∣ z ∣^{2} < c$ (bounded), so it is relatively compact in $Ω$ .

(2) $\Rightarrow$ (1). Suppose $Ω$ has a continuous PSH exhaustion $φ$ . We must show $- lo g δ$ is PSH. The key is the disc-continuity criterion: $- lo g δ$ is PSH if and only if for every analytic disc $f : \overline{D} \to Ω$ (continuous on $\overline{D}$ , holomorphic on $D$ , with values in $C^{n}$ ) one has $- lo g δ (f (0)) \leq sup_{∣ ζ ∣ = 1} (- lo g δ (f (ζ)))$ , equivalently $δ (f (0)) \geq in f_{∣ ζ ∣ = 1} δ (f (ζ))$ . Fix such a disc and any unit vector $b \in C^{n}$ ; the shifted disc $ζ \mapsto f (ζ) + t b$ stays in $Ω$ for $t$ below $in f_{∣ ζ ∣ = 1} δ (f (ζ))$ , because $φ \circ (f + t b)$ is subharmonic in $ζ$ and bounded on the boundary circle by $sup_{φ}$ over the boundary disc, so by the maximum principle the shifted disc cannot escape the sublevel set its boundary lies in. Letting $t$ increase to the boundary distance over the rim and applying this to the centre gives $δ (f (0)) \geq in f_{∣ ζ ∣ = 1} δ (f (ζ))$ , the disc inequality. Hence $- lo g δ$ is PSH.

(1) $\Rightarrow$ (3). Let $p \in \partial Ω$ and $w \in T_{p}^{C} \partial Ω$ , so $\sum_{j} (\partial ρ / \partial z_{j}) (p) w_{j} = 0$ . In the Lemma's identity the middle term vanishes at the boundary in the limit, leaving

j, k \sum \frac{\partial ^{2} ( - lo g δ )}{\partial z _{j} \partial z ˉ _{k}} (z) w_{j} \overset{w}{ˉ}_{k} = \frac{1}{δ ( z )} L_{ρ} (z; w) + O (1) ∣ w ∣^{2}

as $z \to p$ along the inner normal with $w$ kept in (the nearby) complex tangent space. Plurisubharmonicity of $- lo g δ$ forces the left side $\geq 0$ ; multiplying by $δ (z) \to 0^{+}$ and taking the limit gives $L_{ρ} (p; w) \geq 0$ . As $p, w$ were arbitrary, $Ω$ is Levi pseudoconvex.

(3) $\Rightarrow$ (1). This is the harder direction. Suppose $Ω$ is Levi pseudoconvex. One first shows $- lo g δ$ is PSH on a neighbourhood of $\partial Ω$ inside $Ω$ : by the Lemma, on the complex tangent space the Hessian of $- lo g δ$ is $δ^{- 1} L_{ρ} \geq 0$ up to $O (1)$ , and on the missing one-dimensional normal complement the term $δ^{- 2} ∣ \sum_{j} (\partial ρ / \partial z_{j}) w_{j} ∣^{2}$ dominates, being of order $δ^{- 2}$ against the $O (δ^{- 1})$ corrections; so the full complex Hessian of $- lo g δ$ is positive semidefinite near $\partial Ω$ . Away from $\partial Ω$ one corrects by passing to $φ = - lo g δ + C ∣ z ∣^{2}$ on a compact inner region and gluing (the gluing lemma for PSH functions, 06.10.02 Exercise 6), producing a global continuous PSH exhaustion; then (2) $\Rightarrow$ (1) returns plurisubharmonicity of $- lo g δ$ itself. $□$

Bridge. This equivalence builds toward the solution of the Levi problem and appears again in the Hörmander $L^{2}$ existence theorem for $\overset{ˉ}{\partial}$ , where condition (2) supplies the strictly PSH weight whose Levi form makes the basic estimate close. The foundational reason the three notions coincide is the Lemma: the complex Hessian of the boundary-distance barrier $- lo g δ$ is the Levi form of the boundary divided by the distance, so a local sign condition on $\partial Ω$ and a global convexity condition on the barrier are the same datum measured at two scales. This is exactly the bridge promised in 06.10.01 and 06.10.02: the distance function $- lo g d (z, \partial Ω)$ identifies the geometric boundary form with the global no-leak property. The equivalence generalises holomorphic convexity — replacing the test family $O (Ω)$ of 06.10.01 by the PSH functions of 06.10.02 converts holomorphic convexity into pseudoconvexity — and the central insight is that putting these together yields the easy half of the Levi problem: every domain of holomorphy is pseudoconvex, proved next.

Exercises Intermediate+

Exercise 3 (medium, short-answer).

Show that the polydisc $P (0, 1) = {∣ z_{j} ∣ < 1, j = 1, \dots, n}$ is pseudoconvex by exhibiting a continuous PSH exhaustion, without computing any Levi form (the boundary is only Lipschitz, not $C^{2}$ ).

Hint

The functions $- lo g (1 - ∣ z_{j} ∣^{2})$ are PSH on the polydisc; combine them.

Answer

For each $j$ , $- lo g (1 - ∣ z_{j} ∣^{2})$ is PSH on $P (0, 1)$ : it is a function of $z_{j}$ alone equal to $- lo g$ of $(1 - ∣ z_{j} ∣^{2})$ , and $- lo g (1 - ∣ λ ∣^{2})$ is subharmonic (indeed convex increasing in $∣ λ ∣^{2}$ ) on the one-variable disc, so it slices to subharmonic in every direction. The sum $φ (z) = \sum_{j = 1}^{n} - lo g (1 - ∣ z_{j} ∣^{2})$ is then PSH, continuous, and an exhaustion: $φ < c$ forces each $∣ z_{j} ∣^{2} < 1 - e^{- c} / n \cdot (\dots)$ , in any case each $∣ z_{j} ∣$ bounded away from $1$ , so ${φ < c} ⋐ P (0, 1)$ . By the equivalence theorem, the polydisc is pseudoconvex. Rubric: full credit for the PSH sum and the exhaustion property; the boundary-distance form $- lo g δ$ equals $- lo g min_{j} (1 - ∣ z_{j} ∣)$ and is also PSH.

Exercise 4 (medium, symbolic).

Let $ρ (z) = Re (z_{n}) + ∣ z_{1} ∣^{2} + \dots + ∣ z_{n - 1} ∣^{2}$ near $0$ , defining a domain ${ρ < 0}$ . Compute $T_{0}^{C} \partial Ω$ and the Levi form on it, and classify the point $0$ .

Hint

$Re (z_{n}) = \frac{1}{2} (z_{n} + \overset{z}{ˉ}_{n})$ , so $\partial ρ / \partial z_{n} = 1/2$ and the complex tangent space is ${w_{n} = 0}$ .

Answer

Here $\partial ρ / \partial z_{n} (0) = 1/2$ and $\partial ρ / \partial z_{j} (0) = \overset{z}{ˉ}_{j} ∣_{0} = 0$ for $j < n$ , so $T_{0}^{C} \partial Ω = {w : \frac{1}{2} w_{n} = 0} = {w_{n} = 0}$ , of complex dimension $n - 1$ . The complex Hessian is $\partial^{2} ρ / \partial z_{j} \partial \overset{z}{ˉ}_{k} = δ_{j k}$ for $j, k \leq n - 1$ and $0$ if either index is $n$ . On $T_{0}^{C}$ (where $w_{n} = 0$ ), $L_{ρ} (0; w) = \sum_{j = 1}^{n - 1} ∣ w_{j} ∣^{2} > 0$ for nonzero $w$ . So $0$ is a strictly pseudoconvex point; this $ρ$ is the local model (the Siegel/paraboloid normal form) of a strictly pseudoconvex boundary. Rubric: full credit for $T_{0}^{C} = {w_{n} = 0}$ and the strictly positive Levi form there.

Exercise 6 (hard, short-answer).

Prove that an increasing union $Ω = ⋃_{j} Ω_{j}$ of pseudoconvex domains with $Ω_{j} \subseteq Ω_{j + 1}$ is pseudoconvex (Behnke–Stein for pseudoconvexity).

Hint

The boundary distances increase: $δ_{Ω_{j}} \leq δ_{Ω_{j + 1}}$ on $Ω_{j}$ , so $- lo g δ_{Ω_{j}}$ decreases; relate the limit to $- lo g δ_{Ω}$ .

Answer

Fix a point $z_{0} \in Ω$ ; it lies in some $Ω_{j_{0}}$ and in all later $Ω_{j}$ . On any compact $K ⋐ Ω$ , $K \subseteq Ω_{j}$ for large $j$ , and $δ_{Ω_{j}} (z) ↑ δ_{Ω} (z)$ as $j \to \infty$ for each $z \in Ω$ (the regions grow, so the distance to the boundary grows, monotonically), with the convergence locally uniform by Dini since the $δ_{Ω_{j}}$ are increasing continuous and the limit $δ_{Ω}$ is continuous. Hence $- lo g δ_{Ω_{j}} ↓ - lo g δ_{Ω}$ pointwise on $Ω$ . Each $- lo g δ_{Ω_{j}}$ is PSH on $Ω_{j}$ ; restricting to a fixed compact and using that a decreasing limit of PSH functions is PSH (06.10.02 Exercise 5), $- lo g δ_{Ω}$ is PSH on every compact, hence on $Ω$ . So $Ω$ is pseudoconvex. This is the pseudoconvex analogue of the holomorphic-convexity Behnke–Stein result of 06.10.01. Rubric: full credit for the monotone increase of $δ_{Ω_{j}}$ and the decreasing-PSH-limit argument.

Exercise 7 (hard, short-answer).

Prove the easy half of the Levi problem: every domain of holomorphy $Ω \subseteq C^{n}$ is pseudoconvex. Use the Kontinuitätssatz (continuity principle) for analytic discs.

Hint

If $- lo g δ$ failed to be PSH, a family of analytic discs would push their centres toward the boundary while their boundaries stay away from it; a domain of holomorphy forbids this.

Answer

Suppose $Ω$ is a domain of holomorphy. We verify the disc inequality $δ (f (0)) \geq in f_{∣ ζ ∣ = 1} δ (f (ζ))$ for every analytic disc $f : \overline{D} \to Ω$ , which is equivalent to $- lo g δ \in PSH (Ω)$ . Suppose not: there is a disc with $δ (f (0)) < in f_{∣ ζ ∣ = 1} δ (f (ζ)) =: 2 r$ . Translate the disc family $f_{t} = f + t b$ in some direction $b$ ; the boundary circles $f_{t} (\partial D)$ stay at distance $> r$ from $\partial Ω$ for $∣ t ∣ \leq r$ , forming a compact family $S = ⋃_{∣ t ∣ \leq r} f_{t} (\partial D) ⋐ Ω$ , while some centre $f_{t_{0}} (0)$ reaches $\partial Ω$ . By the maximum principle, every $g \in O (Ω)$ satisfies $sup_{f_{t} (\overline{D})} ∣ g ∣ = sup_{f_{t} (\partial D)} ∣ g ∣ \leq sup_{S} ∣ g ∣$ , so the centres $f_{t} (0)$ all lie in the holomorphically convex hull $\hat{S}_{Ω}$ of 06.10.01. But a domain of holomorphy is holomorphically convex (Cartan–Thullen, 06.10.01), so $\hat{S}_{Ω} ⋐ Ω$ , contradicting a centre reaching $\partial Ω$ . Hence the disc inequality holds and $Ω$ is pseudoconvex. Rubric: full credit for the disc-translation family, the hull containment via the maximum principle, and the Cartan–Thullen contradiction.

Advanced results Master

Pseudoconvexity is the geometric face of the structure theory begun in 06.10.01 and 06.10.02. The results below sharpen the Levi-form picture, record the bumping and exhaustion technology that the $\overset{ˉ}{\partial}$ method consumes, and state the full Levi problem whose easy half was proved above.

Smoothing and strict exhaustions. A pseudoconvex $Ω$ admits not merely a continuous PSH exhaustion but a smooth strictly PSH exhaustion. Starting from $φ_{0} = max (- lo g δ, ∣ z ∣^{2})$ , one regularises by the convolution of 06.10.02 on each shell, glues the smoothings across shells with the PSH gluing lemma, and adds a rapidly growing convex function of $∣ z ∣^{2}$ to upgrade semidefiniteness to definiteness. The resulting $φ \in C^{\infty} (Ω)$ with $L_{φ} > 0$ everywhere is the weight that the Hörmander $L^{2}$ existence theorem requires; the smooth strictly PSH exhaustion is, in effect, the analytic incarnation of pseudoconvexity.

Levi-flat boundaries and the foliation. When the Levi form vanishes identically on $T^{C} \partial Ω$ along a boundary piece, that piece is Levi-flat. The complex tangent spaces $T_{p}^{C} \partial Ω$ then integrate to a foliation of the boundary by complex hypersurfaces — the Levi foliation. The model is ${Re (z_{n}) = 0}$ , foliated by the affine hyperplanes ${z_{n} = i t}$ . Levi-flat hypersurfaces are the boundary between strict pseudoconvexity and pseudoconcavity and are the source of the worst regularity behaviour for the $\overset{ˉ}{\partial}$ -Neumann problem.

Bumping and local-to-global. Strict pseudoconvexity is stable and localisable: near a strictly pseudoconvex boundary point one can bump the domain outward, enlarging $Ω$ to a strictly larger pseudoconvex $\tilde{Ω}$ that agrees with $Ω$ away from the point, by adding a small multiple of a peak function $∣ z - p ∣^{2} - ε$ to the defining function. This bumping is the mechanism behind Oka's solution of the Levi problem on strongly pseudoconvex domains: one solves $\overset{ˉ}{\partial}$ on the slightly larger domain and restricts. Weakly pseudoconvex points admit no such universal bump, which is why the general Levi problem needed the global $L^{2}$ method rather than a local kernel construction.

The Levi problem. The converse of the easy half — every pseudoconvex domain is a domain of holomorphy — is the Levi problem proper, solved by Oka for $n = 2$ in 1942 and in general by Oka, Bremermann, and Norguet in 1953–1954, and re-proved analytically by Hörmander's 1965 weighted $L^{2}$ estimates for $\overset{ˉ}{\partial}$ . Combined with Cartan–Thullen (06.10.01) and the PSH-hull identity (06.10.02), it closes the circle: for a domain $Ω \subseteq C^{n}$ ,

domain of holomorphy ⟺ holomorphically convex ⟺ pseudoconvex ⟺ PSH-exhaustible ⟺ H^{q \geq 1} (Ω, O) = 0.

The Levi form sits at the boundary-geometry end of this chain; the cohomological vanishing sits at the analytic end; and $- lo g δ$ is the function that ties them.

Synthesis. Pseudoconvexity is the foundational reason the structure theory of $C^{n}$ has a local-geometric criterion at all: the Levi form converts the global no-leak condition of 06.10.01 into a pointwise sign condition on the boundary, and this is exactly the same datum as the plurisubharmonicity of the barrier $- lo g δ$ , read at the scale $δ \to 0$ . Putting these together, the geometric notion (Levi form $\geq 0$ on the complex tangent space), the potential-theoretic notion (continuous PSH exhaustion), and the function-theoretic notion (holomorphic convexity, 06.10.01) are three faces of one structure, and the central insight is that the boundary-distance height $- lo g d (z, \partial Ω)$ is the universal certificate identifying all three. The easy half of the Levi problem is dual to Cartan–Thullen: where Cartan–Thullen produces a singular function from a non-compact hull, the disc-continuity argument produces a hull point from a failure of plurisubharmonicity, and the bridge is in both cases the holomorphically convex hull. This pattern recurs throughout the chapter: the smooth strictly PSH exhaustion built here is the weight on which the Hörmander $L^{2}$ theory runs, and the strict Levi form is the positivity that makes its basic estimate close.

Full proof set Master

Proposition 1 (the Levi form on $T_{p}^{C}$ is independent of the defining function). Let $ρ_{1}, ρ_{2}$ be two $C^{2}$ defining functions for $Ω$ near $p \in \partial Ω$ . Then $L_{ρ_{1}} (p; w)$ and $L_{ρ_{2}} (p; w)$ are positive multiples of each other for $w \in T_{p}^{C} \partial Ω$ ; in particular the sign of the Levi form on the complex tangent space is well defined.

Proof. Two defining functions for the same boundary near $p$ satisfy $ρ_{2} = h ρ_{1}$ with $h$ a positive $C^{1}$ function (where $ρ_{1} \neq = 0$ , and $h = ∣\nabla ρ_{2} ∣/∣\nabla ρ_{1} ∣ > 0$ on ${ρ_{1} = 0}$ by comparing gradients of two functions with the same zero set and nonvanishing differential). Compute the complex Hessian by the product rule:

\frac{\partial ^{2} ρ _{2}}{\partial z _{j} \partial z ˉ _{k}} = h \frac{\partial ^{2} ρ _{1}}{\partial z _{j} \partial z ˉ _{k}} + \frac{\partial h}{\partial z _{j}} \frac{\partial ρ _{1}}{\partial z ˉ _{k}} + \frac{\partial h}{\partial z ˉ _{k}} \frac{\partial ρ _{1}}{\partial z _{j}} + ρ_{1} \frac{\partial ^{2} h}{\partial z _{j} \partial z ˉ _{k}} .

Evaluate at $p$ (so $ρ_{1} (p) = 0$ , killing the last term) and contract against $w \otimes \overset{w}{ˉ}$ with $w \in T_{p}^{C} \partial Ω$ , i.e. $\sum_{j} (\partial ρ_{1} / \partial z_{j}) (p) w_{j} = 0$ and its conjugate $\sum_{k} (\partial ρ_{1} / \partial \overset{z}{ˉ}_{k}) (p) \overset{w}{ˉ}_{k} = 0$ . The two cross terms vanish because each contracts a derivative of $ρ_{1}$ against $w$ or $\overset{w}{ˉ}$ in a complex-tangent direction. What survives is $L_{ρ_{2}} (p; w) = h (p) L_{ρ_{1}} (p; w)$ with $h (p) > 0$ . $□$

Proposition 2 (strict pseudoconvexity is an open, biholomorphically invariant condition). Let $Ω$ have $C^{2}$ boundary. The set of strictly pseudoconvex boundary points is open in $\partial Ω$ , and strict pseudoconvexity at $p$ is preserved by any biholomorphism defined near $p$ .

Proof. Openness: $L_{ρ} (p; w)$ depends continuously on $p$ , and positive-definiteness of a continuously varying Hermitian form on the continuously varying subspace $T_{p}^{C} \partial Ω$ is an open condition (the minimal eigenvalue over the unit sphere of the tangent space is continuous and positive at $p$ , hence positive nearby). Invariance: let $F$ be biholomorphic near $p$ with $F (p) = q$ , and let $σ$ be a defining function for $F (Ω)$ near $q$ , so $ρ = σ \circ F$ defines $Ω$ near $p$ . Since $F$ is holomorphic, $\partial F_{m} / \partial \overset{z}{ˉ}_{k} = 0$ , and the chain rule gives

\frac{\partial ^{2} ρ}{\partial z _{j} \partial z ˉ _{k}} (p) = m, l \sum \frac{\partial ^{2} σ}{\partial ζ _{m} \partial ζ ˉ _{l}} (q) \frac{\partial F _{m}}{\partial z _{j}} (p) \overline{\frac{\partial F _{l}}{\partial z _{k}} (p)} + m \sum \frac{\partial σ}{\partial ζ _{m}} (q) \frac{\partial ^{2} F _{m}}{\partial z _{j} \partial z ˉ _{k}} (p) .

The second sum vanishes because each $F_{m}$ is holomorphic, so $\partial^{2} F_{m} / \partial z_{j} \partial \overset{z}{ˉ}_{k} = 0$ . Hence $L_{ρ} (p; w) = L_{σ} (q; F^{'} (p) w)$ , where $F^{'} (p)$ is the holomorphic Jacobian, and $F^{'} (p)$ maps $T_{p}^{C} \partial Ω$ isomorphically onto $T_{q}^{C} \partial F (Ω)$ . Positive-definiteness of $L_{σ}$ on the latter therefore transfers to positive-definiteness of $L_{ρ}$ on the former. $□$

Proposition 3 (a strictly pseudoconvex point is locally biholomorphic to a strictly convex one). If $\partial Ω$ is strictly Levi pseudoconvex at $p$ , there is a local biholomorphism $F$ near $p$ such that $F (Ω)$ is strictly convex near $F (p)$ .

Proof. Choose holomorphic coordinates centred at $p$ in which the defining function has the second-order Taylor expansion

ρ (z) = Re (z_{n}) + j, k \sum \frac{\partial ^{2} ρ}{\partial z _{j} \partial z ˉ _{k}} (0) z_{j} \overset{z}{ˉ}_{k} + Re (j, k \sum \frac{\partial ^{2} ρ}{\partial z _{j} \partial z _{k}} (0) z_{j} z_{k}) + o (∣ z ∣^{2}),

after the linear change putting $\partial ρ (0)$ in the $z_{n}$ -direction. The holomorphic change of variable $w_{n} = z_{n} + \sum_{j, k} (\partial^{2} ρ / \partial z_{j} \partial z_{k}) (0) z_{j} z_{k}$ , $w_{j} = z_{j}$ ( $j < n$ ) absorbs the holomorphic Hessian term $Re (\sum z_{j} z_{k} \partial^{2} ρ / \partial z_{j} \partial z_{k})$ into $Re (w_{n})$ . In the new coordinates the second-order part is $Re (w_{n}) + L_{ρ} (0; w)$ with the Levi form positive definite on the complex tangent ${w_{n} = 0}$ . Adding $C (Re w_{n})^{2}$ for large $C$ (a further $C^{2}$ change of defining function not affecting the zero set to second order) makes the real Hessian positive definite on the whole real tangent hyperplane, so $F (Ω) = {w : new ρ < 0}$ is strictly convex near $0$ . $□$

Proposition 4 (continuity principle obstructs leakage). Let $Ω$ be pseudoconvex and ${f_{t} : \overline{D} \to C^{n}}_{t \in [0, 1]}$ a continuous family of analytic discs with $⋃_{t} f_{t} (\partial D) ⋐ Ω$ and $f_{0} (\overline{D}) \subseteq Ω$ . Then $f_{t} (\overline{D}) \subseteq Ω$ for all $t$ .

Proof. Let $φ$ be a continuous PSH exhaustion of $Ω$ (Theorem, condition (2)). The boundary union $S = ⋃_{t} f_{t} (\partial D)$ is compact in $Ω$ , so $M := sup_{S} φ < \infty$ . For each $t$ with $f_{t} (\overline{D}) \subseteq Ω$ , the composition $φ \circ f_{t}$ is subharmonic on $D$ and continuous on $\overline{D}$ , so by the maximum principle $sup_{\overline{D}} φ \circ f_{t} = sup_{\partial D} φ \circ f_{t} \leq M$ ; thus $f_{t} (\overline{D})$ lies in the sublevel set ${φ \leq M} ⋐ Ω$ . Let $T = {t : f_{t} (\overline{D}) \subseteq Ω}$ . It contains $0$ , is open (a disc in the open set ${φ < M + 1} ⋐ Ω$ stays in $Ω$ under small perturbation), and is closed: if $t_{n} \to t$ with $f_{t_{n}} (\overline{D}) \subseteq {φ \leq M}$ , the limit disc $f_{t} (\overline{D})$ lies in the compact ${φ \leq M} ⋐ Ω$ by continuity. So $T = [0, 1]$ . This is the precise sense in which pseudoconvexity forbids the disc-leakage that destroys a domain of holomorphy, and it is the engine of Exercise 7. $□$

Connections Master

Plurisubharmonic functions 06.10.02. This unit is the geometric payoff of the PSH theory: pseudoconvexity is defined through the plurisubharmonicity of $- lo g d (z, \partial Ω)$ , and the equivalence with a continuous PSH exhaustion is proved using the maximum principle, the max-stability, and the decreasing-limit and gluing lemmas established there. The Levi form here is the boundary restriction of the complex Hessian that defined plurisubharmonicity.
Domains of holomorphy and holomorphic convexity 06.10.01. The easy half of the Levi problem proved in Exercise 7 and Proposition 4 shows every domain of holomorphy is pseudoconvex, using Cartan–Thullen and the holomorphically convex hull directly. Pseudoconvexity is the local-geometric reformulation of the global holomorphic convexity characterised there; the full Levi problem is the converse, completing the equivalence.
Holomorphic functions of several variables 06.07.01. The defining function, the complex tangent space, and the Wirtinger derivatives $\partial / \partial z_{j}$ , $\partial / \partial \overset{z}{ˉ}_{k}$ that build the Levi form are the multi-variable calculus of that unit. The vanishing of the second sum in Propositions 2 and 3 is exactly the holomorphy condition $\partial F_{m} / \partial \overset{z}{ˉ}_{k} = 0$ , so the Levi form's biholomorphic invariance is a genuinely complex-analytic fact with no real-variable shadow.
Dirichlet problem and the Perron method 06.01.24. The disc-continuity criterion and the subharmonicity of $φ \circ f$ on a complex line reduce pseudoconvexity to one-variable subharmonic potential theory; the maximum principle that drives Proposition 4 is the several-variable lift of the one-variable maximum principle developed there.
The $\overset{ˉ}{\partial}$ problem with $L^{2}$ estimates 06.04.05. The smooth strictly PSH exhaustion constructed in the Advanced results is the weight on which the Hörmander weighted $L^{2}$ existence theorem runs; the strict Levi form is the positive curvature term in the Bochner–Kodaira basic estimate, and the solution of the full Levi problem is what that estimate ultimately delivers. The dimension-one $\overset{ˉ}{\partial}$ -Hilbert theory there is the seed of this several-variable machinery.
Survey of Cartan–Serre Stein theory in higher dimensions 06.09.08. Pseudoconvexity is the manifold-level hypothesis under which a complex manifold is Stein; the equivalence chain of the Advanced results is the open-domain case of the general Stein theory surveyed there, with the Levi form supplying the boundary criterion for relatively compact pseudoconvex pieces.

Historical & philosophical context Master

The boundary positivity condition now bearing his name was found by Eugenio Elia Levi in 1910 in Studii sui punti singolari essenziali delle funzioni analitiche di due o più variabili complesse ^{[Levi 1910]} (Annali di Matematica Pura ed Applicata (3) 17, 61–87). Levi analysed the boundaries across which a holomorphic function of two variables cannot be continued and isolated the second-order condition — the nonnegativity of what is now the Levi form on the complex tangent space — that the boundary of a domain of holomorphy must satisfy. He proved the necessity (the easy half) and conjectured the sufficiency, which became the Levi problem.

The reframing that made the problem tractable was the introduction of plurisubharmonic functions and the recognition that the right global object is not the boundary form alone but the function $- lo g d (z, \partial Ω)$ on the whole domain. Kiyoshi Oka, in the sixth of his memoirs Sur les fonctions analytiques de plusieurs variables ^{[Oka 1942]} (Tôhoku Mathematical Journal 49, 15–52), worked with the class he called fonctions pseudoconvexes and established the equivalence of pseudoconvexity with the existence of a plurisubharmonic exhaustion, then solved the Levi problem for $n = 2$ . The general solution followed in 1953–1954 through Oka's ninth memoir and the independent work of Hans-Joachim Bremermann and François Norguet, and Lars Hörmander's 1965 $L^{2}$ estimates for $\overset{ˉ}{\partial}$ supplied a uniform analytic proof valid in all dimensions.

The local-to-global passage in Proposition 3 — that a strictly pseudoconvex boundary point is biholomorphic to a strictly convex one — is the device by which Oka reduced the strongly pseudoconvex case to convexity, where domains of holomorphy were already understood; the weakly pseudoconvex case, where no such local convexification exists, is what forced the global potential-theoretic and PDE methods. Pseudoconvexity thereby occupies the centre of the subject: it is the boundary-geometric condition that Levi found, the potential-theoretic condition that Oka and Lelong formalised, and the function-theoretic condition that Cartan and Thullen characterised, identified through the single barrier $- lo g d (z, \partial Ω)$ .

Bibliography Master

@article{Levi1910,
  author  = {Levi, Eugenio Elia},
  title   = {Studii sui punti singolari essenziali delle funzioni
             analitiche di due o pi\`u variabili complesse},
  journal = {Ann. Mat. Pura Appl. (3)},
  volume  = {17},
  year    = {1910},
  pages   = {61--87}
}

@article{Oka1942,
  author  = {Oka, Kiyoshi},
  title   = {Sur les fonctions analytiques de plusieurs variables. VI.
             {Domaines} pseudoconvexes},
  journal = {T\^ohoku Math. J.},
  volume  = {49},
  year    = {1942},
  pages   = {15--52}
}

@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{Norguet1954,
  author  = {Norguet, Fran\c{c}ois},
  title   = {Sur les domaines d'holomorphie des fonctions uniformes de
             plusieurs variables complexes (passage du local au global)},
  journal = {Bull. Soc. Math. France},
  volume  = {82},
  year    = {1954},
  pages   = {137--159}
}

@article{Bremermann1954,
  author  = {Bremermann, Hans-Joachim},
  title   = {\"Uber die \"Aquivalenz der pseudokonvexen Gebiete und der
             Holomorphiegebiete im Raum von $n$ komplexen
             Ver\"anderlichen},
  journal = {Math. Ann.},
  volume  = {128},
  year    = {1954},
  pages   = {63--91}
}

@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}
}

@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}
}

Prerequisites

06.10.02
06.10.01
06.07.01
06.01.24
06.01.01

Tier anchors

beginner: Krantz *Function Theory of Several Complex Variables* Ch. 2, informal sections; Range *Holomorphic Functions and Integral Representations* Ch. III intro
intermediate: Krantz *Function Theory of Several Complex Variables* §2.6; Hörmander *An Introduction to Complex Analysis in Several Variables* §2.6
master: Levi 1910 / Oka 1942 (originators); Krantz Ch. 2 §2.6; Hörmander §2.6; Range Ch. III

References

Levi 1910 — Studii sui punti singolari essenziali delle funzioni analitiche di due o più variabili complesse · Ann. Mat. Pura Appl. (3) 17, 61–87 (originator of the boundary positivity condition now called the Levi form)
Oka 1942 — Sur les fonctions analytiques de plusieurs variables, VI: Domaines pseudoconvexes · Tôhoku Math. J. 49, 15–52 (independent originator; equates pseudoconvexity with PSH-exhaustibility)
Krantz — Function Theory of Several Complex Variables · AMS Chelsea 340, 2nd ed., Ch. 2 §2.6 (the Levi form, Levi pseudoconvexity, Hartogs pseudoconvexity)
Hörmander — An Introduction to Complex Analysis in Several Variables · North-Holland, 3rd ed. 1990, §2.6 (pseudoconvexity, equivalence with PSH exhaustion functions)
Range — Holomorphic Functions and Integral Representations in Several Complex Variables · Springer GTM 108, Ch. III (Levi problem, equivalence of pseudoconvexity notions)

Estimated time

beginner: 16m
intermediate: 42m
master: 68m