Equiaffine immersions and pseudo-Riemannian space forms

We introduce an explicit construction that produces immersions into the pseudosphere $\mathbb{S}^{n,n+1}$ and the pseudohyperbolic space $\mathbb{H}^{n+1,n}$ starting from equiaffine immersions in $\mathbb{R}^{n+1}$, and conversely. We describe how these immersions interact with a para-Sasaki metric defined on $\mathbb{H}^{n+1,n}$ via a principal $\mathbb{R}$-bundle structure over a para-Kähler manifold. In the case where the immersion in $\mathbb{R}^{n+1}$ is an $n$-dimensional hyperbolic affine sphere, we obtain spacelike maximal immersions in $\mathbb{H}^{n+1,n}$ that satisfy a transversality condition with respect to the principal $\mathbb{R}$-bundle structure. As a first application, we show that, given a certain boundary set $Λ_Ω\subset \partial_\infty \mathbb{H}^{n+1,n}$, associated with a properly convex subset $Ω\subset \mathbb{RP}^n$ and homeomorphic to an $(n-1)$-sphere, there exists an $n$-dimensional maximal spacelike submanifold in $\mathbb{H}^{n+1,n}$ whose boundary is precisely $Λ_Ω$. As a second application, we show that the Blaschke lift of the hyperbolic affine sphere, introduced by Labourie for $n=2$, into the symmetric space of $\mathrm{SL}(n+1,\mathbb{R})$ is a harmonic map.

💡 Research Summary

This paper establishes an explicit and bidirectional correspondence between equiaffine immersions into Euclidean space and immersions into pseudo-Riemannian space forms of neutral signature, leading to significant applications in boundary value problems and harmonic map theory.

The core construction starts from a non-degenerate equiaffine immersion (f, ξ) of an n-dimensional manifold M into R^{n+1}. Using the conormal map ν of this immersion, the authors define two maps: σ_- = (-ξ, ν) and σ_+ = (ξ, ν). These maps naturally land in the space V = R^{n+1} ⊕ (R^{n+1})*, which is equipped with a bilinear form of signature (n+1, n+1). By construction, σ_- has constant norm -1 and thus defines an immersion into the pseudohyperbolic space H^{n+1,n}, while σ_+ has norm +1 and immerses into the pseudosphere S^{n,n+1}.

A key geometric structure on H^{n+1,n} is its principal R-bundle structure over the para-complex hyperbolic space H^n_τ, a para-Kähler manifold. The paper meticulously analyzes how the immersions σ_± interact with this structure and the induced para-Sasaki metric on the total space. The main theorem (Theorem A) shows that for a non-degenerate equiaffine immersion, σ_- is a horizontal and φ-anti-invariant immersion whose induced metric coincides with the affine metric of the dual immersion ν. Its projection σ̄ = π ∘ σ_- is a Lagrangian immersion into H^n_τ.

Crucially, the correspondence translates geometric properties between the affine and pseudo-Riemannian settings. Corollary B establishes the equivalence of the following: σ_- being a maximal (zero mean curvature) submanifold in H^{n+1,n}; σ̄ being maximal in H^n_τ; and the conormal map ν being a proper affine sphere. This link is made explicit via the Pick tensor of ν.

The paper also solves the inverse problem (Theorems C and D): given a simply connected M and a non-degenerate Lagrangian immersion σ̄ into H^n_τ, there exists a unique (up to an additive constant) horizontal lift σ_- to H^{n+1,n}. The components of this lift, ξ and ν, are then centroaffine immersions dual to each other, and they define an equiaffine immersion f. If σ̄ is maximal, then f and ν are proper affine spheres.

The first major application (Theorem E) concerns a Dirichlet-type problem at infinity for the pseudo-hyperbolic space. Given a properly convex set Ω in RP^n, one associates a “hyperplane boundary set” Λ_Ω in the ideal boundary ∂∞H^{n+1,n}. The paper proves the existence of an n-dimensional, spacelike, maximal, horizontal, and φ-anti-invariant submanifold in H^{n+1,n} whose asymptotic boundary is precisely Λ_Ω. The proof leverages the celebrated existence theorem for complete hyperbolic affine spheres asymptotic to the cone over Ω (by Cheng-Yau, etc.) and applies the forward correspondence. The induced metric is complete, and the submanifold is unique up to the natural R-action on H^{n+1,n}. This result is novel because the dimension n of the maximal submanifold differs from the positive index (n+1) of the ambient space form, contrasting with most existing literature.

The second application (Theorem F) generalizes Labourie’s “Blaschke lift” to all dimensions. For a hyperbolic affine sphere f: M → R^{n+1}, the authors define a map G_f into the symmetric space X_{n+1} = SL(n+1, R)/SO(n+1). They prove that G_f is a harmonic map. The strategy involves expressing G_f as a composition of the map σ_+ constructed earlier and another carefully designed map into an intermediate space, then analyzing harmonicity properties under a pseudo-Riemannian submersion. As a corollary, composing with a standard inclusion yields a harmonic map into the symmetric space of SO₀(n+1, n+1).

Overall, the paper provides a powerful and explicit bridge connecting affine differential geometry, pseudo-Riemannian geometry of neutral signature, convex projective geometry, and harmonic map theory, yielding new existence results and generalizing known constructions.