The Möbius characterizations of Willmore tori and Veronese submanifolds in the unit sphere

Zhen Guo, Haizhong Li and Changping Wang

Vol. 241 (2009), No. 2, 227–242

DOI: 10.2140/pjm.2009.241.227

Abstract

Suppose M is a m-dimensional submanifold without umbilic points in the (m + p)-dimensional unit sphere S^m+p. Four basic invariants of M^m under the Möbius transformation group of S^m+p are a symmetric positive definite 2-form g called the Möbius metric, a section B of the normal bundle called the Möbius second fundamental form, a 1-form Φ called the Möbius form, and a symmetric (0,2) tensor A called the Blaschke tensor. In the Möbius geometry of submanifolds, the most important examples of Möbius minimal submanifolds (also called Willmore submanifolds) are Willmore tori and Veronese submanifolds. In this paper, several fundamental inequalities of the Möbius geometry of submanifolds are established and the Möbius characterizations of Willmore tori and Veronese submanifolds are presented by using Möbius invariants.

Keywords

Willmore tori, Veronese submanifolds, Möbius geometry of submanifolds

Mathematical Subject Classification 2000

Primary: 53A30

Secondary: 53B25

Milestones

Received: 8 January 2009

Accepted: 16 January 2009

Published: 1 June 2009