Fine structure of the space of spherical minimal immersions

Abstract

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