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On Locally Strongly Convex Affine Hyperspheres Realizing Chen’s Equality

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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 \(m\le 5\).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471023, China

    Cece Li

  2. College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao, 266590, China

    Cece Li

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Supported by National Natural Science Foundation of China (Grant No. 11401173), and Natural Science Foundation of Henan.

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Li, C. On Locally Strongly Convex Affine Hyperspheres Realizing Chen’s Equality. Results Math 75, 35 (2020). https://doi.org/10.1007/s00025-020-1160-2

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