Fine structure of the space of spherical minimal immersions

Abstract:

The space of congruence classes of full spherical minimal immersions $f:S^m\to S^n$ of a given source dimension $m$ and algebraic degree $p$ is a compact convex body $\mathcal {M}_m^p$ in a representation space $\mathcal {F}_m^p$ of the special orthogonal group $SO(m+1)$. In Ann. of Math. 93 (1971), 43–62 DoCarmo and Wallach gave a lower bound for $\mathcal {F}_m^p$ and conjectured that the estimate was sharp. Toth resolved this “exact dimension conjecture” positively so that all irreducible components of $\mathcal {F}_m^p$ became known. The purpose of the present paper is to characterize each irreducible component $V$ of $\mathcal {F}_m^p$ in terms of the spherical minimal immersions represented by the slice $V\cap \mathcal {M}_m^p$. Using this geometric insight, the recent examples of DeTurck and Ziller are located within $\mathcal {M}_m^p$.