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 such surfaces represent a non-trivial cobordism class together with an observation of M. Gromov relating the conformal area in $\mathbb{S}^n$ to minimal fillings in the hyperbolic space $\mathbb{H}^{n+1}$.
Let $M^n$ be a closed minimal submanifold in the unit sphere $\mathbb{S}^{n+q}$ with $n\geqslant 3$ and $q\geqslant 2$. Let $S$ be the squared length of its second fundamental form and $S_{\max}=\max_{p\in M}S(p)$. We prove that if $M$ is not totally geodesic, then \[ S_{\max}>\frac{2n}{3}+\frac{n-2}{182}. \]
Let $F:M^n\to\mathbb{S}^{2n+1}$ be a minimal Legendrian immersion of a connected closed manifold, where $n\ge3$, and let $B$ be its second fundamental form. We prove that if $|B|^2\le(n-1)(n+2)/n$, then, up to a unitary transformation, $F$ parametrizes a totally geodesic sphere or is a finite covering of the image of t...
We show that for an embedded two-sided minimal hypersurface in $\mathbb{R}^N$, there is a lower bound for the index in terms of the first Betti number and the number of ends, using ideas from the recent work of Chodosh--Gianocca. This estimate is sharp for the higher-dimensional catenoid. We also obtain a $\operatornam...
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...
We prove that a local minimal hypersurface $M^3$ in the unit sphere $S^4$ with constant scalar curvature and three distinct principal curvatures at every point must be locally congruent to the minimal Cartan isoparametric hypersurface in $S^4$, which provides a positive answer to the Bryant conjecture \cite{ChangLocal}...
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...
Jacob Bernstein, Daniel Ketover· 1 citation
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