Scalar invariants of surfaces in the conformal 3-sphere via Minkowski spacetime

Jie Qing, Changping Wang and Jingyang Zhong

Vol. 286 (2017), No. 1, 153–190

DOI: 10.2140/pjm.2017.286.153

Abstract

For a surface in the 3-sphere, by identifying the conformal round 3-sphere as the projectivized positive light cone in Minkowski 5-spacetime, we use the conformal Gauss map and the conformal transform to construct the associate homogeneous 4-surface in Minkowski 5-spacetime. We then derive the local fundamental theorem for a surface in the conformal round 3-sphere from that of the associate 4-surface in Minkowski 5-spacetime. More importantly, following an idea of Fefferman and Graham, we construct local scalar invariants for a surface in the conformal round 3-sphere. One distinct feature of our construction is to link the classic work of Blaschke to the work of Bryant and Fefferman and Graham.

Keywords

scalar invariant, conformal geometry, conformal Gauss map, Willmore surface

Mathematical Subject Classification 2010

Primary: 53A30, 53B25

Milestones

Received: 19 February 2016

Accepted: 29 May 2016

Published: 9 December 2016