The totally geodesic 2-sphere and the Clifford torus in $S^3$ are the only compact, minimal surfaces in $S^3$ whose quadric representations yield minimally in hyperquadrics.
"Spherical Minimal Surfaces with Minimal Quadric Representation in Some Hyperquadric." Tokyo J. Math. 17 (2) 479 - 493, December 1994. https://doi.org/10.3836/tjm/1270127968