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In page 15 of the article New Applications of Min-max Theory, Andre Neves said that: a wishful thinking would suggest the Lawson minimal surface $\xi_{1,2}$ in the 3-sphere to have a Morse index 9, but there is no real evidence.

My question is, what suggests the index to be 9? Say, can we find a 9-parameter deformation of $\xi_{1,2}$ which decreases its area?

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Due to a recent preprint by Kapouleas and Wiygul (arXiv:1904.05812), the index of the Lawson $\xi_{1,g}$ surface is $2g+3$. I have not looked at the paper in detail so far, but let me cite from the authors introduction:

"The ideas of our proof originate with work of NK on the approximate kernel for Scherk surfaces. Our approach requires a detailed understanding of the elementary geometry of $S^3$ and of the surfaces involved, especially their symmetries. The proof makes also heavy use of Alexandrov reflection in the style of Schoen’s. The Courant nodal theorem [ and an argument of Montiel- Ros play important roles as well. In ongoing work we hope to extend this result to determine the index and nullity of all Lawson surfaces desingularizing intersecting great two-spheres in the round three-sphere $S^3$."

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  • $\begingroup$ Wow! A great answer! $\endgroup$ – A. Chu Apr 12 '19 at 16:37

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