Degree 3 algebraic minimal surfaces in the 3-sphere

TL;DR

This paper characterizes degree 3 algebraic minimal surfaces in the 3-sphere, showing they are Lawson's minimal tori, providing an alternative proof to a known classification result.

Contribution

It offers a local analytic characterization of degree 3 algebraic minimal surfaces in the 3-sphere, confirming their identity as Lawson's minimal tori with a new proof approach.

Findings

Degree 3 algebraic minimal surfaces in S^3 are Lawson's minimal tori

Provides an alternative proof of Perdomo's classification

Characterization is based on local analytic properties

Abstract

We give a local analytic characterization that a minimal surface in the 3-sphere $\, \ES^3 \subset \R^4$ defined by an irreducible cubic polynomial is one of the Lawson's minimal tori. This provides an alternative proof of the result by Perdomo (\emph{Characterization of order 3 algebraic immersed minimal surfaces of $S^3$},Geom. Dedicata 129 (2007), 23--34).