On the symmetric equiaffine hyperspheres and the minimal symmetric Lagrangian submanifolds

TL;DR

This paper establishes a duality correspondence between symmetric equiaffine hyperspheres and minimal symmetric Lagrangian submanifolds, providing an alternative proof for their classification with parallel Fubini-Pick forms.

Contribution

It introduces a duality-based approach linking equiaffine hyperspheres and Lagrangian submanifolds, offering a new proof for their classification.

Findings

Established a duality correspondence between the two geometric objects.

Provided an alternative proof for the classification theorem.

Enhanced understanding of the structure of symmetric equiaffine hyperspheres.

Abstract

In this paper, a correspondence via duality is established between the set of locally strongly convex symmetric equiaffine hyperspheres and the set of minimal symmetric Lagrangian submanifolds in a certain complex space form. By using this correspondence theorem, we are able to provide an alternative proof of the classification theorem for the locally strongly convex equiaffine hypersurfaces with parallel Fubini-Pick forms, which has been established recently by Z.J. Hu etc in a totally different way.