Constructing bounded pseudoconvex domains of finite D'Angelo type with prescribed weak loci

Abstract: We construct smoothly bounded pseudoconvex or convex domains in $\mathbb{C}<sup>n$ whose weakly pseudoconvex loci realize prescribed closed sets.

• The paper presents explicit constructions of bounded pseudoconvex domains with prescribed weak loci that maintain finite D'Angelo type.
• It employs Whitney extension, degenerate Monge-Ampère equations, and regularized patching techniques to achieve precise control over the domain’s geometry.
• The work reveals that smooth weak loci can exhibit nonanalytic structures, impacting subelliptic estimates and challenging classical real-analytic constraints.

Constructing Smoothly Bounded Pseudoconvex Domains of Finite D'Angelo Type with Prescribed Weak Loci

Introduction

This paper establishes explicit constructions of smoothly bounded pseudoconvex (and also convex) domains in $\mathbb{C}^n$ whose weakly pseudoconvex loci are arbitrary prescribed closed subsets of $\mathbb{R}^{n-1} \subset \mathbb{C}^{n-1}$. The framework provides robust control on the D'Angelo type along these loci, ensuring finite type, and demonstrates that smooth weak loci may exhibit highly non-analytic structure, in stark contrast with the real-analytic setting. Moreover, the methods developed exploit advanced local and global analysis, including the degenerate real and complex Monge-Ampère equations, as well as geometric patching and regularization techniques ensuring convexity and finite type.

Background and Problem Motivation

The D'Angelo type is central in several complex variables, measuring the maximal order of contact between holomorphic curves and boundaries of domains. Finite D'Angelo type is strictly required for both geometric regularity (e.g., no analytic curves contained in the boundary) and analytic phenomena such as subelliptic $\bar\partial$-Neumann estimates. Real-analyticity imposes rigid algebraic restrictions: compact real-analytic varieties cannot contain germs of holomorphic curves, so the weak locus in this setting is always of finite type and has a constrained analytic structure. In the smooth category, this rigidity is lost; the existence or structure of flat points or weak loci is more flexible, but explicit global control is nontrivial.

The weakly pseudoconvex locus—the set of boundary points where the Levi determinant vanishes—is the critical set for the degeneracy of local geometry. Its structure governs phenomena ranging from the failure of Stein neighborhood bases (as in the Diederich–Fornæss worm domain) to the loss of regularity for canonical operators. The paper's main goal is to realize any given compact set as such a weak locus, while maintaining finite D'Angelo type and smooth boundedness.

Main Results and Construction Scheme

The paper proves that given any compact set $E \subset \mathbb{R}^{n-1}$ (or lower-codimension slice for convex models), one can construct a globally bounded smooth pseudoconvex (or convex) domain $\Omega \subset \mathbb{C}^n$ so that the weak locus $W \subset b\Omega$ is diffeomorphic to $E$ (or to a fiber bundle over $E$, depending on the model). Furthermore, the D'Angelo type at points of $W$ is explicitly controlled, with an upper bound $2m$ or $\mathbb{R}^{n-1} \subset \mathbb{C}^{n-1}$0 depending on the precise construction.

The key technical advances include three components:

1. Smooth Potential Construction: The authors use Whitney extension techniques to build a smooth function $\mathbb{R}^{n-1} \subset \mathbb{C}^{n-1}$1 which vanishes precisely on $\mathbb{R}^{n-1} \subset \mathbb{C}^{n-1}$2 and is flat there, ensuring that the potential’s Hessian degenerates exactly along $\mathbb{R}^{n-1} \subset \mathbb{C}^{n-1}$3.
2. Local Solution to Monge-Ampère Equations: In dimensions $\mathbb{R}^{n-1} \subset \mathbb{C}^{n-1}$4, they employ local solvability results for the (real or complex) degenerate Monge–Ampère equation, guaranteeing the existence of smooth (pluri)subharmonic potentials with determinant vanishing precisely on the prescribed set.
3. Regularized Maximum and Patching: Unbounded or local models are globalized by patching with strictly convex or strongly pseudoconvex barriers using a regularized maximum operator: a mollified max function preserving convexity and pseudoconvexity, maintaining smoothness of the boundary, and guaranteeing that no spurious new weak points are introduced in the transition region.

These allow for essentially arbitrary flexibility in the “footprint” of the weak locus, in marked contrast to analytic or even typical generic assumptions in several complex variables.

Notable Models and Results

• For $\mathbb{R}^{n-1} \subset \mathbb{C}^{n-1}$5 and arbitrary compact $\mathbb{R}^{n-1} \subset \mathbb{C}^{n-1}$6, the construction produces a convex domain whose weak locus is $\mathbb{R}^{n-1} \subset \mathbb{C}^{n-1}$7 in the boundary (the “Cantor forest” if $\mathbb{R}^{n-1} \subset \mathbb{C}^{n-1}$8 is a Cantor set). By further control in the “Reeb direction,” the fiber can be collapsed, realizing the weak locus diffeomorphic to $\mathbb{R}^{n-1} \subset \mathbb{C}^{n-1}$9.
• In higher dimensions ($\bar\partial$0), for closed sets in lower-dimensional slices, weaker convexity or pseudoconvexity constraints apply due to the nature of the real Monge-Ampère equation. The weak loci can be prescribed with sharp control of the D'Angelo type, depending on codimension.
• All constructed domains are smoothly bounded, pseudoconvex (or convex), with the Levi determinant vanishing to finite order along $\bar\partial$1. The D'Angelo type bound is precisely quantified (with explicit extremal curves achieving the bound).

Analytical Details and Type Computations

The verification of the order of contact (D'Angelo type) and the Levi degeneracy reduction rely on:

• Rigorous coordinate normalization (affine/linear biholomorphic shifts), which ensure that the nontrivial singularity in the defining function is concentrated in explicitly constructed “flat” directions.
• For each model, the computation checks that the complex tangential direction along $\bar\partial$2 or its product with other real directions indeed produces order of contact $\bar\partial$3 (or $\bar\partial$4), using the algebraic separation of harmonic and non-harmonic terms in the Taylor expansion.
• Convexity is preserved everywhere by analyzing the full real Hessian obtained via the chain rule, and leveraging the positivity of all terms (regularized maximum, convex or plurisubharmonic pieces, and barrier).
• The patching process is shown not to create spurious weak points due to careful quantitative estimates on the supports of local and global defining functions, choice of mollifier parameter, and strict barrier scaling.

Theoretical and Practical Implications

The constructions show that in the smooth (i.e., non-real-analytic) category, the local structure of the weakly pseudoconvex locus is essentially unconstrained: arbitrary compact sets—even highly nonrectifiable or of fractional dimension—can be realized as the locus where the Levi determinant vanishes. This flexibility means that the fine geometry of weak loci, and thus of their associated hypoelliptic and subelliptic PDEs, is much richer than the classical analytic case.

These models have implications for:

• The study of the subelliptic $\bar\partial$5-Neumann problem, as regularity of solutions depends critically on the geometry and measure-theoretic properties of the weak locus.
• The theory of CR singularities, as they offer test cases where maximal degeneracy of the Levi form does not arise from analytic varieties or submanifolds.
• Realization theory in CR geometry and foliation theory, especially for construction of exotic boundary behavior while maintaining analytic control.

The conjectures posed suggest further investigations into fine measure-theoretic structure of weak loci and vanishing properties of the Levi determinant. In particular, the (unproven in $\bar\partial$6) claim that the weakly pseudoconvex locus always has zero surface measure if the D'Angelo type is finite everywhere, and that the Levi determinant never vanishes to infinite order, would imply substantial regularity constraints on seemingly flexible degeneration.

Conclusion

The paper rigorously demonstrates that for every compact (possibly totally disconnected, e.g., Cantor-type) set $\bar\partial$7, there exist smoothly bounded, finitely-typed (i.e., of controlled D'Angelo type) pseudoconvex or convex domains in $\bar\partial$8 such that the weakly pseudoconvex locus matches $\bar\partial$9 (or a specified bundle over $E \subset \mathbb{R}^{n-1}$0). The approach synthesizes advanced analytic and geometric PDE techniques (Whitney extension, degenerate Monge–Ampère potentials, and smoothing barriers) into a constructive framework yielding flexible yet analytically robust global domains. This work fundamentally extends the understanding of the potential complexity and flexibility of degeneracy loci for geometric analysis in several complex variables and CR geometry.

Reference: "Constructing bounded pseudoconvex domains of finite D'Angelo type with prescribed weak loci" (2607.10785).