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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?

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    $\begingroup$ 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.) $\endgroup$ – Sebastian Mar 17 at 15:53
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    $\begingroup$ 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)." $\endgroup$ – Michael Albanese Mar 17 at 22:24

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