Horospherical Convexity (h-convexity) in Hyperbolic Spaces

• Horospherical convexity is a hyperbolic convexity concept where supporting horoballs replace half-spaces and smooth boundaries satisfy curvature conditions (κ_i ≥ 1 or κ_i > 1 for strict cases).
• The horospherical support function and Gauss map enable reparametrization of h-convex hypersurfaces over the sphere, facilitating analysis through fully nonlinear PDEs.
• Variants such as weak and horocyclic convexity extend the concept to immersed hypersurfaces and Hadamard-manifold optimization, linking geometric support with asymptotic structures.

Horospherical convexity, usually abbreviated h-convexity, is a hyperbolic analogue of Euclidean convexity in which affine half-spaces are replaced by horoballs, horospheres, or Busemann-function sublevel sets. In hyperbolic space, a bounded domain $K\subset \mathbb H^{n+1}$ is h-convex if every boundary point admits a supporting horoball; for smooth boundaries this is equivalent to the principal-curvature condition $\kappa_i\ge 1$, while strict or uniform h-convexity requires $\kappa_i>1$ (Luo et al., 2024). Closely related variants appear for immersed hypersurfaces, planar horocyclically convex domains, and Hadamard-manifold optimization, all built from the same principle: the supporting objects are adapted to the asymptotic geometry of nonpositively curved spaces rather than to linear structure (Bonini et al., 2016, Arango et al., 2024, Criscitiello et al., 22 May 2025).

1. Supporting horoballs and curvature thresholds

The basic geometric definition in $\mathbb H^{n+1}$ is a support condition. A bounded domain $K\subset \mathbb H^{n+1}$ is h-convex if every boundary point of $\partial K$ admits a supporting horoball, that is, a horoball $B$ with $K\subset B$ and the boundary point lying on $\partial B$. Geometrically, this means that the domain is an intersection of horoballs, just as a Euclidean convex body is an intersection of half-spaces. In the hyperboloid model, horospheres are the hypersurfaces whose principal curvatures are all equal to $1$, so for a smooth boundary the support condition is equivalent to

$\kappa_i\ge 1$0

and strict h-convexity to

$\kappa_i\ge 1$1

This curvature threshold is the hyperbolic shift that distinguishes horospherical convexity from ordinary geodesic convexity (Luo et al., 2024, Andrews et al., 2017).

A weaker immersion-theoretic version is weak horospherical convexity. For an oriented immersed hypersurface $\kappa_i\ge 1$2, one considers the tangent horosphere $\kappa_i\ge 1$3 at $\kappa_i\ge 1$4 whose outward unit normal agrees with the chosen normal field. The hypersurface is weakly horospherically convex at $\kappa_i\ge 1$5 if a neighborhood of $\kappa_i\ge 1$6 stays on one side of $\kappa_i\ge 1$7, equivalently if all principal curvatures satisfy either

$\kappa_i\ge 1$8

The corresponding uniform version requires a constant $\kappa_i\ge 1$9 with $\kappa_i>1$0 everywhere, and this is the class used in the global correspondence and embeddedness theory (Bonini et al., 2016).

This support viewpoint also has lower-dimensional analogues. In the unit disk $\kappa_i>1$1, a proper subdomain $\kappa_i>1$2 is horocyclically convex if every interior boundary point admits a supporting horodisk. The paper on horocyclically convex domains emphasizes the analogy

$\kappa_i>1$3

and states that every $\kappa_i>1$4-convex domain is horo-convex (Arango et al., 2024).

2. Support functions, Gauss maps, and conformal correspondences

For smooth uniformly h-convex hypersurfaces, horospherical convexity is encoded by a support function on the sphere. If $\kappa_i>1$5 is the position vector of $\kappa_i>1$6 and $\kappa_i>1$7 is the outward unit normal, then

$\kappa_i>1$8

for some $\kappa_i>1$9 and $\mathbb H^{n+1}$0. The function

$\mathbb H^{n+1}$1

is the horospherical support function, and the associated horospherical Gauss map $\mathbb H^{n+1}$2 sends each boundary point to the direction $\mathbb H^{n+1}$3. For uniformly h-convex domains this Gauss map is a diffeomorphism, so the hypersurface can be reparametrized over $\mathbb H^{n+1}$4. In this parametrization the decisive tensor is

$\mathbb H^{n+1}$5

and uniform h-convexity is equivalent to

$\mathbb H^{n+1}$6

Moreover,

$\mathbb H^{n+1}$7

so the eigenvalues of $\mathbb H^{n+1}$8 are the hyperbolic curvature radii

$\mathbb H^{n+1}$9

The horospherical surface area measure is then

$K\subset \mathbb H^{n+1}$0

and the $K\subset \mathbb H^{n+1}$1-th horospherical $K\subset \mathbb H^{n+1}$2-surface area measure is

$K\subset \mathbb H^{n+1}$3

These formulas make h-convexity directly usable in fully nonlinear PDE (Luo et al., 2024).

A parallel formulation exists for weakly horospherically convex hypersurfaces via the light-cone map. For an oriented immersion $K\subset \mathbb H^{n+1}$4 with unit normal $K\subset \mathbb H^{n+1}$5, the light-cone map is

$K\subset \mathbb H^{n+1}$6

If $K\subset \mathbb H^{n+1}$7 is the hyperbolic Gauss map, then

$K\subset \mathbb H^{n+1}$8

where $K\subset \mathbb H^{n+1}$9 is the horospherical support function, and the induced horospherical metric is

$\partial K$0

When $\partial K$1 is injective, this becomes a conformal metric $\partial K$2 on a domain $\partial K$3. The local correspondence is governed by the tensor

$\partial K$4

whose eigenvalues \