# Is every minimal hypersurface in $S^n$ algebraic?

Let $$S^n$$ be the round n-sphere. Wu-yi Hsiang asked in his paper “Remarks on closed minimal submanifolds in the standard riemannian m-sphere” (1967) the follow question

Is every minimal hypersurface (i.e. codimensional one) in $$S^n$$ algebraic (i.e. given by intersecting the zero set in $$R^{n+1}$$ of some homogenous polynomial in n+1 variables with $$S^n$$) ?

e.g. the Lawson surface in $$S^3$$ is given by the imaginary part of $$(x_1+ix_2)^m(x_3+ix_4)^l$$.

What is the current status of this problem?

• It should be emphazized that in your question you mean the Lawson surfaces $\tau_{m,k}$ and not the (more famous) Lawson surfaces $\xi_{m,k}$. In fact, Lawson conjectured that $\xi_{2,2}$ is not algebraic (page 350 in 'Complete Minimal Surfaces in S3' Lawson, Annals of Math, Vol. 92, No. 3, 335-374.) – Sebastian Mar 17 at 15:53
• I asked Lawson about this before I saw Sebastian's comment. He still does not know if $\xi_{2,2}$ is algebraic or not, but had the following to say: "There are surfaces constructed in that paper with the property that they are not invariant under $x \mapsto -x$ in $\mathbb{R}^4$. So if they are the zeros of a homogeneous polynomial, they are not all the zeros. Taking minus the surface gives another geometric component of the algebraic variety (if it were algebraic)." – Michael Albanese Mar 17 at 22:24