Minimal surfaces in the round n-sphere are prominent examples of surfaces critical

for the Willmore bending energy W; those of low area provide candidates for Wminimizers.

To understand when such surfaces are W-stable, we study the

interplay between the spectra of their Laplace-Beltrami, area-Jacobi and W-Jacobi

operators. We use this to prove: 1) the square Clifford torus in the 3-sphere is the

only W-minimizer among minimal 2-tori in the n-sphere for all n≥3, evidence for

the higher-codimension Willmore Conjecture (unresolved by Marques and Neves);

2) the hexagonal Itoh-Montiel-Ros torus in the 5-sphere is the only other W-stable

minimal 2-torus in the n-sphere for all n≥5; 3) the Itoh-Montiel-Ros torus is a local

minimum for the conformally-constrained Willmore problem, evidence for a recent

conjecture of Heller and Pedit. We can also give sharp estimates on the Morse

index for the area of minimal 2-tori in the n-sphere.