Scalar Invariants of surfaces in conformal 3-sphere via Minkowski spacetime

TL;DR

This paper develops a method to construct scalar invariants of surfaces in the conformal 3-sphere by leveraging Minkowski spacetime, linking classical differential geometry with conformal invariants through a novel geometric approach.

Contribution

It introduces a new construction of scalar invariants for surfaces in conformal 3-sphere using Minkowski spacetime and connects classical Blaschke invariants with modern conformal geometry.

Findings

Derived local fundamental theorem for conformal surfaces

Constructed scalar invariants linking Blaschke and Fefferman-Graham theories

Established a geometric framework connecting classical and modern conformal invariants

Abstract

For a surface in 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 conformal round 3-sphere from that of the associate 4-surface in Minkowski 5-spacetime. More importantly, following the idea of Fefferman and Graham, we construct local scalar invariants for a surface in conformal round 3-sphere. One distinct feature of our construction is to link the classic work of Blaschke to the works of Bryan and Fefferman-Graham.