Minimal tori with low nullity

The nullity of a minimal submanifold M ⊂ S n is the dimension of the nullspace of the second variation of the area functional. That space contains as a subspace the effect of the group of rigid motions SO ( n + 1 ) of the ambient space, modulo those motions which preserve M, whose dimension is the Killing nullity k n ( M ) of M. In the case of 2-dimensional tori M in S 3 , there is an additional naturally-defined 2-dimensional subspace that contributes to the nullity; the dimension of the sum of the action of the rigid motions and this space is the natural nullity n n t ( M ) . In this paper we will study minimal tori in S 3 with natural nullity less than 8. We construct minimal immersions of the plane R 2 in S 3 that contain all possible examples of tori with n n t ( M ) < 8 . We prove that the examples of Lawson and Hsiang with k n ( M ) = 5 also have n n t ( M ) = 5 , and we prove that if the n n t ( M ) ⩽ 6 then the group of isometries of M is not trivial.