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

Abstract

For each integer $k \geq 3$ Hermann Karcher identified a complete singly periodic minimal surface $Ξ_k$ (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 $Ξ_{k,m}$ for the quotient of $Ξ_k$ by translation through $m \geq 1$ fundamental periods, we study the Morse index and nullity of the subfamilies $Ξ_{k,2}$ and $Ξ_{3,m}$. In the $m=2$ case we prove for all $k \geq 3$ that $Ξ_{k,2}$ has Morse index $4k-3$ and nullity $3$. In the $k=3$ case we prove that there exists a real number $α^* \in (1/3,1/2)$ such that for all $m \geq 1$ the Morse index of $Ξ_{3,m}$ is $6m- 4 \lfloor mα^* \rfloor - 3$ and its nullity is $3$ unless $mα^*$ 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).