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We show that among M\"obius bands in $\mathbb{S}^3$ bounded by a great circle, the minimal Willmore energy is realized by an embedded minimal M\"obius band with Morse index two. To prove this, we introduce a $2$-parameter ``canonical family"associated to any non-orientable surface in $\mathbb{S}^3$ with boundary a grea...
We show that a great circle in $\mathbb{S}^3$ bounds embedded non-orientable minimal surfaces of unbounded genera and all possible non-zero Euler numbers. This answers a question of R. Hardt and H. Rosenberg (1990). It also gives the first example of a real analytic Jordan curve bounding infinitely many embedded minima...
For each $n\geq 4$, we show that the infimal Willmore energy of a non-orientable surface of odd genus immersed in $\mathbb{S}^n$ is attained by an index $(n+1)$ minimal surface in this class. For $n=4$, we obtain in addition the analogous conclusion for surfaces with non-zero Euler number. The proof uses the fact that...
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