On the index of minimal 2-tori in the 4-sphere

Abstract

Abstract In this note, we prove that any minimal 2-torus in S 4 S^{4} has Morse index at least 6, with equality if and only if it is congruent to the Clifford torus in some great S 3 ⊂ S 4 S^{3}\subset S^{4} . For a minimal 2-torus in S n S^{n} with vanishing Hopf differential, we show that its index is at least n + 3 n+3 and that this estimate is sharp: the equilateral 2-torus fully embedded in S 5 ⊂ S n S^{5}\subset S^{n} as a homogeneous minimal surface in S n S^{n} has index exactly n + 3 n+3 .