Characterization of order 3 algebraic immersed minimal surfaces of S3

In this paper we prove that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi M\to S^3$$\end{document} is a minimal immersion of a compact surface and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi(M)= S^3\cap f^{-1}(0)$$\end{document} , for some homogeneous polynomial f of degree 3 on R4, then, M is a torus and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi(M)$$\end{document} is one of the examples given by Lawson (1970, Complete minimal surfaces in S3. Ann. Math. 92(2), 335–374).