Strictly Pseudo-Convex Domains in Cn

• Strictly pseudo-convex domains in Cn are domains defined by a C2 function whose Levi form is positive on the boundary, ensuring both local and global analytic regularity.
• They play a central role in solving the ∂̅-problem and extending holomorphic functions, which is critical for the development of function theory on complex manifolds.
• Recent advances extend their analysis to Banach spaces and projective settings, revealing phenomena like unique boundary behavior and non-pluripolar core properties.

A strictly pseudo-convex domain in $\mathbb{C}^n$ is a fundamental object in several complex variables and complex geometry, distinguished by the positivity of the Levi form of a defining function for the domain at every boundary point. Such domains play a central role in the analytic and geometric theory of complex manifolds, providing the natural setting for the solution of the $\bar{\partial}$-problem, for the extension of holomorphic functions, and for the development of function theory analogous to classical domains in $\mathbb{C}$. The following sections develop a comprehensive account of the theory, characterizations, geometric implications, analytic properties, and recent advances concerning strictly pseudo-convex domains in $\mathbb{C}^n$.

1. Geometric and Analytic Characterization

A domain $D\subset\mathbb{C}^n$ is (strictly) pseudo-convex if there exists a $C^2$ defining function $\rho$ in a neighborhood of $\overline{D}$ (i.e., $D = \{z\in U : \rho(z) < 0\}$, with $d\rho \neq 0$ on $bD$) such that the complex Hessian (the Levi form)

$$\mathcal{L}_\rho(p; X) = \sum_{i,j=1}^n \frac{\partial^2 \rho}{\partial z_i \partial \bar z_j}(p) X_i \overline{X_j}$$

is positive definite on the complex tangent space $T^{\mathbb{C}}_p(bD)$ for each $p \in bD$. Thus, strict pseudo-convexity aligns with the uniform positivity of the Levi form across the boundary. The analytic content of this condition is twofold:

• Existence of a strictly plurisubharmonic function $\varphi$ defining $D$ near the boundary, i.e., $\varphi$ is $C^2$, strictly plurisubharmonic in a neighborhood $U$ of $\overline{D}$, and $D = \{\varphi < 0\}$ with $d\varphi\neq 0$ on $bD$ (Harz et al., 2014).
• The property is stable under biholomorphic mappings and, for such domains, one can always choose a global strictly plurisubharmonic defining function in the ambient complex manifold.

This strict positivity property is not only local but reflects deep global analytic implications via the construction of smooth exhaustion functions and the availability of powerful $L^2$ methods for holomorphic extension and regularity.

2. Rigid and Localization-Based Characterizations

A pivotal result concerns the "rigid" characterization of pseudo-convexity via balanced domains and fibers (Nikolov et al., 2010). For any open $D\subset\mathbb{C}^n$, pseudoconvexity is equivalent to the following:

• For every $z\in D$, the largest balanced domain

$$B_{D,z} := z + I_{D,z}, \quad I_{D,z} := \{ X\in\mathbb{C}^n : z + \lambda X \in D\ \forall |\lambda|<1\}$$

is pseudoconvex.

• The associated Hartogs-like domain $$H_D = \{(z,w)\in D\times\mathbb{C}^n : w\in I_{D,z}\}$$ is pseudoconvex.

Furthermore, the plurisubharmonicity of $-\log(dp(z, X))$, where $dp(z,X)$ is the distance from $z$ to $bD$ in direction $X$, characterizes pseudoconvexity. For linearly convex (in the sense of supporting complex hyperplanes) domains, analogues exist: local convexity of the balanced fibers reflects global convexity (Nikolov et al., 2010). Slicing techniques show that, particularly for $n\ge 3$, the pseudoconvexity of all $2$-dimensional complex planes through a boundary point encodes global pseudoconvexity.

These equivalences considerably strengthen the understanding of local-to-global phenomena: strictly pseudoconvex domains, by virtue of the positivity of the Levi form, are rigid under such localization.

3. Boundary Regularity, Exposing Points, and Finite Type

Boundary regularity and finite type conditions sharpen the scope of strictly pseudoconvex domains (Diederich et al., 2013). For a bounded domain $\Omega\subset \mathbb{C}^n$ that is locally convexifiable near a boundary point $p$ of finite 1-type $2k$, with a Stein neighborhood basis, there exists a biholomorphism $f:\overline{\Omega}\to \mathbb{C}^n$ exposing $p$ as an extreme point of type $2k$. Explicitly, $f$ maps $p$ to a point in $\mathbb{C}^n$ such that local geometry near $p$ is modeled by domains of the form

$$B_k = \{ z \in \mathbb{C}^n : |z_n|^2 + |z'|^{2k} < 1\}$$

where $|z'|$ corresponds to the $(n-1)$-components. This realization confirms that strictly pseudoconvex points (the $k=1$ case) admit global biholomorphic normal forms, which is crucial for constructing support functions, studying peak point theory, and establishing boundary regularity of the Bergman and Szegő kernels. The methods make extensive use of exposing maps, holomorphic support functions, and parameter-dependent automorphism theory (Andersen–Lempert).

4. Metric and Dynamical Properties

The strictly pseudoconvex condition ensures that intrinsic complex-geometric metrics exhibit strong curvature properties. For instance, strictly pseudoconvex bounded domains endowed with the Kobayashi distance are Gromov hyperbolic; that is, geodesic triangles are $\delta$-thin, and global negative curvature analogues hold (Gaussier et al., 2013). The absence of analytic disks in the boundary is also a necessary condition for Gromov hyperbolicity, while the presence of analytic disks (even in convex domains) destroys this property. Notably, certain weakly pseudoconvex, smoothly bounded convex domains (e.g., complex ellipsoids with boundaries of finite type) achieve Gromov hyperbolicity if analytic disks are absent.

A closely related property, visibility (the requirement that near-geodesics joining near-boundary points intersect a fixed compact set), is satisfied by strictly pseudoconvex domains, and it is shown that for domains with $C^2$-smooth boundaries, visibility implies pseudoconvexity (Nikolov et al., 3 Jul 2024).

5. Analytic Properties: $\bar{\partial}$-Problem and $L^r$ Estimates

Strictly pseudoconvex domains are the premier context for $L^2$ and $L^p$ regularity theories of the inhomogeneous Cauchy–Riemann equations: $\bar{\partial}u = \omega, \quad \omega \in L^r.$ Large classes of $L^r$–$L^s$ estimates are available (Amar, 2013). For $1 < r < 2(n+1)$,

$$\|u\|_{L^s} \leq C\|\omega\|_{L^r}, \qquad \frac{1}{s} = \frac{1}{r} - \frac{1}{2(n+1)}$$

and for critical and supercritical exponents, endpoint estimates in BMO or Lipschitz spaces. The analysis uses Carleson measure techniques and subordination lemmas involving the lifting of functions to auxiliary strictly pseudoconvex domains, facilitating access to Skoda-type estimates.

The closed range property for $\bar{\partial}$ holds on strictly pseudoconvex domains, supporting the existence and regularity of the $\bar{\partial}$-Neumann operator and enabling canonical solution operators in $L^2$ (Herbig et al., 2014).

6. Extensions, Cores, and Non-Classical Phenomena

Every strictly pseudoconvex domain $\Omega$ with smooth boundary in a complex manifold $\mathcal{M}$ admits a global defining function $\varphi$ that is strictly plurisubharmonic near $b\Omega$ (Harz et al., 2014). However, the global structure may be influenced by the presence of a "core" $\mathfrak{c}(\Omega)$, the set of points where all smooth bounded above plurisubharmonic functions fail to be strictly plurisubharmonic. Remarkably, examples exist of strictly pseudoconvex domains whose core has non-empty interior and is therefore not pluripolar (Slodkowski, 2021).

Recent results address domains in projective spaces, establishing that—under suitable topological and convexity conditions—strictly pseudoconvex domains with $C^2$ boundaries admit strictly plurisubharmonic (psh) functions in neighborhoods of the boundary, thereby linking analytic and topological data (Sibony, 2020). Additionally, for partial pseudoconvexity, the union (exhaustion) by strictly pseudoconvex domains preserves the property (Hu et al., 23 Mar 2025).

Non-classical phenomena emerge as well: for instance, there exist strictly pseudoconvex domains in $\mathbb{C}^2$ whose squeezing function (measuring how "ball-like" a domain is) is not plurisubharmonic (Fornæss et al., 2016), and bounded convex domains with $C^2$ boundary for which the squeezing function tends to $1$ near the boundary, yet the domain is not strictly pseudoconvex (Fornæss et al., 2016). These counterexamples refine the boundaries of classical expectation regarding metric and function theory in such domains.

7. Generalizations and Further Directions

The theory has been generalized to broader settings, including Banach spaces, where new notions such as $\ell$-strict and $\ell$-uniform pseudoconvexity, and strong pseudoconvexity, capture the curvature and analytic properties without requiring $C^2$ smoothness (Castillo, 2017). This extension is nontrivial and necessitates the deployment of distributional and averaging approaches to plurisubharmonicity, and it remains compatible with solvability of the $\bar{\partial}$-equation and boundary regularity in infinite-dimensional contexts.

For strictly $q$-pseudoconvex domains, the theory links geometric convexity conditions (eigenvalue count of the Levi form) with function theory, showing that strictly $q$-pseudoconvex domains are globally $(q+1)$-holomorphically convex and, under further assumptions, may exhibit even stronger $q$-holomorphic convexity (Ionita et al., 2018).

Biholomorphic invariants, boundary rigidity (as captured by the obstruction function in the Fefferman–Cheng–Yau expansion), and the structure of curvature tensors in CR geometry (e.g., the vanishing of the CR obstruction implies local sphericity for boundaries in dimension $3$) have deep connections to strictly pseudoconvex domains, especially in $\mathbb{C}^2$ (Curry et al., 2018, Son, 2015).

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In summary, strictly pseudoconvex domains in $\mathbb{C}^n$ are distinguished by the Levi positivity of their smooth defining function, inducing rich analytic, geometric, and metric structures. Recent advances elucidate rigid characterizations, analytic regularity, metric hyperbolicity, boundary behavior, and generalizations to infinite-dimensional and projective settings, while also uncovering significant exceptions and core phenomena that challenge earlier generalities. This intricate interplay underscores their continuing centrality in complex analysis and several complex variables.