Morse index of Karcher saddle towers in ${\mathbb{R}}^2\times {\mathbb{S}}^1(m)$

Abstract

For each integer $k\ge 3$ Hermann Karcher identified a complete singly periodic minimal surface ${\Xi }_k$ (unique up to similarity) with $2k$ ends asymptotic to the union of $k$ planes intersecting equiangularly along a single line and with genus zero in the quotient by a fundamental translation. Writing ${\Xi }_{k,m}$ for the quotient of ${\Xi }_k$ by translation through $m\ge 1$ fundamental periods, we study the Morse index and nullity of the subfamilies ${\Xi }_{k,2}$ and ${\Xi }_{3,m}$. In the $m=2$ case we prove for all $k\ge 3$ that ${\Xi }_{k,2}$ has Morse index $4k-3$ and nullity $3$. In the $k=3$ case we prove that there exists a real number ${\alpha }^{\ast }\in (1/3,1/2)$ such that for all $m\ge 1$ the Morse index of ${\Xi }_{3,m}$ is $6m-4\lfloor m{\alpha }^{\ast }\rfloor -3$ and its nullity is $3$ unless $m{\alpha }^{\ast }$ is an integer, in which case its nullity is $7$. Both proofs exploit the symmetries of each surface and the Dirichlet-to-Neumann map on a half period to reduce the problem to Fourier-analytic computations on the unit circle (after a conformal change).

This version includes minor corrections and expository changes as well as more details (in the acknowledgments) on the role of AI in the project