Free boundary minimal M\"obius band in spherical caps

TL;DR

This paper investigates free boundary minimal M"obius bands in spherical caps, proving their rotational symmetry, constructing explicit examples, and extending Morse index estimates for such submanifolds.

Contribution

It establishes the intrinsic rotational symmetry of these M"obius bands, provides explicit constructions, and generalizes Morse index bounds for free boundary minimal submanifolds.

Findings

Free boundary minimal M"obius bands must be rotationally symmetric.

Explicit construction of such bands in $ ext{B}^4(r)$ for $0<r<rac{ ext{pi}}{2}$.

Non totally geodesic immersions have Morse index at least $n$.

Abstract

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 $n$.