Free boundary minimal M\"obius band in spherical caps

We study compactly free boundary minimal submanifolds in spherical caps $\Br$ and their geometric spectral properties. Following the foundational work of Fraser-Schoen \cite{FS2012}, Lima-Menezes \cite{LM23} established the connection between free boundary minimal surfaces in spherical caps and spectral geometry. In this work, we present three main contributions: (1) We prove that any free boundary minimal M\"obius band in $\Br$ immersed by first Steklov eigenfunctions must be intrinsically rotationally symmetric ; (2) We explicitly construct such a M\"obius band in $\mathbb{B}^4(r)$ for $0<r<\frac{\pi}{2}$; and (3) We generalize Morse index estimates for free boundary minimal submanifolds in spherical caps, showing that non totally geodesic immersions have index at least $n$.