Equiaffine immersions and pseudo-Riemannian space forms

TL;DR

This paper develops a method to construct and analyze immersions into pseudo-Riemannian space forms from equiaffine immersions, revealing new geometric properties and applications such as boundary behavior and harmonicity of certain lifts.

Contribution

It introduces explicit constructions linking equiaffine and pseudo-Riemannian immersions, and explores their geometric and boundary properties, including applications to maximal submanifolds and harmonic maps.

Findings

Constructed immersions into pseudosphere and pseudohyperbolic space from equiaffine immersions.

Established existence of maximal spacelike submanifolds with prescribed boundary sets.

Proved that the Blaschke lift of hyperbolic affine spheres is harmonic.

Abstract

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-K\"ahler 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 $\Lambda_\Omega \subset \partial_\infty \mathbb{H}^{n+1,n}$, associated with a properly convex subset $\Omega \subset \mathbb{RP}^n$ 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 $\Lambda_\Omega$. 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.