Equiaffine immersions and pseudo-Riemannian space forms

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-Kähler manifold ${\mathbb{H}}_{\tau }^n$, called the para-complex hyperbolic space. 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, given a strictly convex subset $\Omega \subset {\mathbb{RP}}^n$, we define a boundary set ${\overline{\Lambda }}_{\Omega }$ in the partial flag variety of lines and hyperplanes in ${\mathbb{R}}^{n+1}$, and prove the existence and uniqueness of a spacelike, Lagrangian, maximal $n$-submanifold in ${\mathbb{H}}_{\tau }^n$ with boundary ${\overline{\Lambda }}_{\Omega }$. We also discuss its implications in the case of ${\mathbb{H}}^{n+1,n}$. 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.

arXiv commentComments welcome! Theorem E updated. Sections 1.2 and 4.1 revised