Regular Affine Domains in Geometric Analysis

• Regular affine domains are convex open sets defined as the interior of intersections of C-null half-spaces in an affine space, generalizing classical domains of dependence.
• They are characterized analytically via solutions to Monge–Ampère equations, yielding unique foliations by smooth hypersurfaces with constant affine Gaussian curvature especially in three dimensions.
• This framework unifies Lorentzian and affine geometries by bridging geometric convexity with analytic properties and by employing tools like Legendre transforms and comparison principles.

A regular affine domain is a precise generalization of the classical domains of dependence in Minkowski space, extending their definition to arbitrary affine spaces associated with proper convex cones. In this formalism, regular affine domains serve as the natural domains for foliation by smooth, locally strongly convex hypersurfaces of constant affine Gaussian curvature. Their characterization synthesizes geometric convexity with deep analytic properties via solutions to certain Monge–Ampère equations with extended-real, lower semicontinuous boundary conditions. In dimension three, each proper regular affine domain admits a unique smooth foliation by such hypersurfaces, providing a compelling geometric-unifying principle across Lorentzian and affine geometries.

1. Definition of Regular Affine Domain

Let $V$ be an $(n+1)$-dimensional real vector space equipped with a fixed volume form, and let $A$ denote the corresponding affine space. A convex cone $C \subset V$ is called proper if it contains no entire affine line. The concept of regularity is anchored in the structure of "supporting half-spaces": a supporting half-space $H$ of $C$ is any open half-space in $V$ whose boundary is a hyperplane through the origin and which contains $C$. Among these, $C$-null half-spaces—those with boundary hyperplanes intersecting $\partial C$ only at the origin—play a critical role.

A $C$-regular domain $D \subset A$ is defined as the interior of an arbitrary intersection of closed $C$-null half-spaces, i.e.,

$D = \operatorname{Int}\left(\bigcap_{H\in\mathcal{H}} \overline{H}\right), \quad \mathcal{H} \subset \{\text{$C$-null half-spaces}\}.$

This construction, when $C$ is the future light cone in Minkowski space $\mathbb{R}^{n,1}$, recovers the familiar domains of dependence or regular domains. The properness of $C$ guarantees that $D$ is a nontrivial convex open set, and the classification into "null" vs. "spacelike" supports aligns the formalism with features of Lorentzian geometry.

2. Analytic Characterization via Monge–Ampère Equations

In an appropriate affine chart, the dual cone $C^{*}$ projects to a bounded convex domain $\Omega \subset \mathbb{R}^n$. Each $C$-regular domain $D$ corresponds bijectively (cf. Theorem 4.3(1)) to a lower-semicontinuous function $$\varphi : \Omega \rightarrow \mathbb{R} \cup \{+\infty\}$$ through the dual (support) transform:

$$D = \left\{ (y,\,\xi) \in \mathbb{R}^{n+1} : \xi > \varphi^*(y) \right\},$$

where $\varphi^*$ is the Legendre transform of $\varphi$.

Selecting a canonical $C$-null boundary-defining function is facilitated by solving an analytically intricate Dirichlet-type Monge–Ampère equation. The archetypal Cheng–Yau equation for hyperbolic affine spheres reads:

$$\det D^2 w = (-w)^{-n-2} \quad \text{on } \Omega, \quad w|_{\partial\Omega} = 0,$$

yielding the support function $w_\Omega$. For surfaces of constant affine Gauss–Kronecker curvature $k>0$, the equation is extended (see Corollary 6.4):

$$\det D^2 u(x) = c_k(-w_\Omega(x))^{-n-2}, \quad x \in U := \operatorname{Int}\operatorname{Dom} u, \quad u|_{\partial \Omega} = \varphi, \quad |\nabla u(x)| \to +\infty \text{ as } x \to \partial U,$$

where $\varphi: \Omega \to \mathbb{R} \cup \{+\infty\}$ is the lower semicontinuous boundary datum, $u \in LC(\mathbb{R}^n)$ is convex and finite precisely on $U$, and $c_k > 0$ depends on $k$ and $n$.

3. Existence and Uniqueness of the Dirichlet Problem

For $n=2$, existence and uniqueness of convex solutions to the Dirichlet problem above are ensured under a mild exterior-circle (or strict convexity) hypothesis on $\Omega$ and provided the lower-semicontinuous function $\varphi$ has nonempty effective domain. Specifically, there exists a unique $u \in LC(\mathbb{R}^2)$, finite precisely on $U \subset \Omega$, satisfying:

$$\det D^2 u = c\,(-w_\Omega)^{-4} \text{ in } U, \quad u|_{\partial\Omega} = \varphi, \quad |\nabla u| \to \infty \text{ on } \partial U.$$

The arguments invoke the Aleksandrov–Heinz strict convexity theorem, Evans–Krylov regularity, and a generalized comparison principle, yielding existence, uniqueness, and interior smoothness.

In higher dimensions ($n \geq 3$), the classical Dirichlet problem for the Monge–Ampère equation requires elevated boundary regularity for well-posedness; when both $\Omega$ and $\varphi$ are $C^\infty$, unique solvability follows from Li–Simon–Chen. At minimal regularity, smooth solutions may fail to exist.

4. Foliation by Constant Affine Gaussian Curvature Hypersurfaces (Dimension 3)

In $\mathbb{R}^3$, the Monge–Ampère solution $u$ defines a convex hypersurface $\Sigma \subset \mathbb{R}^3$ as the graph of $u^*$, which is necessarily spacelike and complete with respect to the cone $C$.

Affine Gauss–Kronecker curvature is defined as follows: Let $\Sigma \subset A$ be a locally strongly convex hypersurface with Blaschke normal $N$, with $h_{ij}$ denoting the affine metric (first fundamental form) and the affine second fundamental form. Then

$$K_{\mathrm{aff}}(\Sigma) = \frac{\det(h_{ij})}{\det(g_{ij})},$$

where $g_{ij}$ refers to the first fundamental form or affine metric, subject to normalization conventions; equivalently, $K_{\mathrm{aff}} = \det S$, for $S$ the shape operator.

For each proper $C$-regular domain $D \subset \mathbb{R}^3$ satisfying a projective exterior-circle condition on $P(C^*)$ and every $k>0$, there exists a unique complete affine $(C,k)$-surface $\Sigma_k \subset D$ asymptotic to $\partial D$. The family $\{\Sigma_k\}_{k>0}$ foliates $D$, and the associated "time-function" $K(p) = \log k$ on $\Sigma_k$ is convex on $D$.

5. Geometric Significance and Relation to Classical Theories

When $C$ is the light cone in Minkowski space, the resulting CAGC (constant affine Gaussian curvature) hypersurfaces correspond precisely to the classical surfaces of constant Gaussian curvature in $\mathbb{R}^{2,1}$, reconstructing the foliation of domains of dependence by space-like slices at constant curvature, as established by Bonsante–Smillie–Seppi and BBZ.

When $D$ is the cone $C$ itself, the foliation is by homothetic Cheng–Yau affine spheres, with the defining equation $\det D^2 K = a e^{b K}$. This recovers the hyperbolic-affine-sphere foliation paradigm developed by Cheng–Yau for proper cones.

The described framework unifies and generalizes both the Lorentzian and affine paradigms: one can deform the canonical cone-foliation by affine spheres in arbitrary directions $\varphi \in LC(\Omega)$, and every deformed domain retains a canonical convex "time-function" $K$ by solving a two-step Monge–Ampère problem.

6. Notable Examples and Special Cases

Several explicit instances elucidate the general theory:

• For the standard Lorentz cone $C_0 \subset \mathbb{R}^{2,1}$, $\Omega$ is the unit disk, $w_\Omega(x) = -\sqrt{1 - |x|^2}$, and the PDE becomes $\det D^2 u = (1 - |x|^2)^{-2}$. The unique solution $u_t$ parameterizes the hyperboloid foliation $\Sigma_k$ of the light cone.
• For simplicial cones ("Titeica affine spheres"), if $C$ is the positive span of a unimodular basis $\{v_0, \dots, v_n\}$, then $\Omega$ is a simplex $\Delta$, $w_\Delta(x) \propto -(t_0 \cdots t_n)^{1/(n+1)}$, and the explicit foliation by constant CAGC surfaces can be constructed.
• For triangular cones in $\mathbb{R}^3$, the existence of a unique affine $(C, k)$-foliation is equivalent to the projective shape of $C$ (with a circumscribed triangular cone $T$) satisfying an exterior-circle condition; failure of the circle condition at a vertex precludes the existence of a complete CAGC surface generating $T$.

7. Summary and Unifying Perspective

Regular affine domains in affine space $A^n$ associated with a proper convex cone $C$ are precisely the interiors of intersections of $C$-null half-spaces. Such domains are characterized analytically via solutions to a two-step Monge–Ampère equation with extended-real, lower semicontinuous boundary data. In three dimensions, every proper regular affine domain admits a canonical, unique foliation by smooth, locally strongly convex hypersurfaces of constant affine Gaussian curvature, thereby unifying classical results on affine spheres and constant-Gaussian-curvature slices in flat Lorentzian spaces (Nie et al., 2019).