Willmore Stability of Minimal Surfaces in Spheres

Rob Kusner, University of Massachusetts at Amherst
Fine Hall 110

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 W-minimizers.  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 2-tori in the n-sphere; 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; 3) the Itoh-Montiel-Ros torus is a local minimum for the conformally-constrained Willmore problem, evidence for a recent conjecture of Lynn Heller and Franz Pedit.  We also give sharp estimates on the Morse index of the area for minimal 2-tori in the n-sphere.