Minimal tori with low nullity

The nullity of a minimal submanifold M⊂Sn 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 kn(M) of M. In the case of 2-dimensional tori M in S3, 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 nnt(M). In this paper we will study minimal tori in S3 with natural nullity less than 8. We construct minimal immersions of the plane R2 in S3 that contain all possible examples of tori with nnt(M)<8. We prove that the examples of Lawson and Hsiang with kn(M)=5 also have nnt(M)=5, and we prove that if the nnt(M)⩽6 then the group of isometries of M is not trivial.