On the index of the critical M\"obius band in $\mathbb B^4$

. In this paper we prove that the Morse index of the critical M¨obius band in the 4 − dimensional Euclidean ball B 4 equals 5. It is conjectured that this is the only embedded non-orientable free boundary minimal surface of index 5 in 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 B 3 equals 4.

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