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

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 $g \geq 2$. 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.

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Nikolaos Kapouleas was partially supported by NSF grant DMS-1405537.

Received 23 May 2019

Published 21 April 2020