The index and nullity of the Lawson surfaces ${\xi }_{g,1}$

Abstract

We prove that the Lawson surface ${\xi }_{g,1}$ in Lawson's original notation, which has genus $g$ and can be viewed as a desingularization of two orthogonal great two-spheres in the round three-sphere ${\mathbb{S}}^3$, has index $2g+3$ and nullity $6$ for any genus $g\ge 2$. In particular ${\xi }_{g,1}$ has no exceptional Jacobi fields, which means that it cannot `flap its wings' at the linearized level and is $C^1$-isolated.

28 pages, no figures. Various typos corrected and minor improvements introduced