On Locally Strongly Convex Affine Hyperspheres Realizing Chen's Equality

摘要

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 Dm of dimension m for some 2 <= m <= n. The present author and Xu (J Math Anal Appl 456:1495-1516, 2017) proposed a conjecture for 2 <= m <= 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 Dm. In particular, we confirm the conjecture for m <= 5.