# The index and nullity of the Lawson surfaces $\xi_{g,1}$

## TL;DR

This paper calculates the index and nullity of Lawson surfaces 1, showing they are linearized isolated with no exceptional Jacobi fields, and provides explicit formulas for these spectral invariants for all genus g.

## Contribution

It provides explicit formulas for the index and nullity of Lawson surfaces 1, demonstrating their linearized stability and isolation for all genus g.

## Key findings

- Index of 1 is 2g+3 for all g2.
- Nullity of 1 is 6 for all g2.
- 1 surfaces have no exceptional Jacobi fields, indicating linearized isolation.

## 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 $g\ge2$. 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.