We aim at the Willlmore conjecture in higher co-dimension. It is natural to ask whether the Clifford torus is Willmore stable when the co-dimension increases and whether there are other Willmore stable tori or not.
We answer these problems for minimal surfaces in $S^n$, by showing that the Clifford torus in $S^3$ and the equilateral Itoh--Montiel--Ros torus in $S^5$ are the only Willmore stable minimal tori in arbitrary higher co-dimension. Moreover, the Clifford torus is the only minimal torus (locally) minimizing the Willmore energy in arbitrary higher codimension. And the equilateral Bryant--Itoh--Montiel--Ros torus is a constrained-Willmore (local) minimizer, but not a Willmore (local) minimizer.
We also generalize Urbano's Theorem to minimal tori in $S^4$ by showing that a minimal torus in $S^4$ has index at least $6$ and the equality holds if and only if it is the Clifford torus. This is a joint work with Prof. Rob Kusner (UMass Amherst).