On the index of the critical Möbius band in ${\mathbb{B}}^4$

Abstract

In this paper we prove that the Morse index of the critical Möbius band in the $4-$dimensional Euclidean ball ${\mathbb{B}}^4$ equals 5. It is conjectured that this is the only embedded non-orientable free boundary minimal surface of index 5 in ${\mathbb{B}}^4$. One of the ingredients in the proof is a comparison theorem between the spectral index of the Steklov problem and the energy index. The latter also enables us to give another proof of the well-known result that the index of the critical catenoid in ${\mathbb{B}}^3$ equals 4.

arXiv comment25 pages. To appear in The Journal of Geometric Analysis