On Locally Strongly Convex Affine Hyperspheres Realizing Chen’s Equality

Abstract

In this paper, we study the locally strongly convex affine hyperspheres realizing Chen’s equality. On such affine hyperspheres of dimension n being not hyperquadrics, there naturally exists a canonical integrable distribution $${\mathbb {D}}_m$$ of dimension m for some $$2\le m\le n$$. The present author and Xu (J Math Anal Appl 456:1495–1516, 2017) proposed a conjecture for $$2\le m\le n-1$$ and a problem for $$m=n$$ to classify them, where the conjecture was confirmed when $$m=2, 3$$, and the problem was solved for 3-dimensional proper affine hyperspheres. As main results, we prove the conjecture under a natural condition of $${\mathbb {D}}_m$$. In particular, we confirm the conjecture for $$m\le 5$$.