19 problems

For every $n,m\ge 1$, the set of possible constant values of $S=\lvert h\rvert^2$ for closed, minimally immersed $n$-dimensional submanifolds $M^n\hookrightarrow \mathbb{S}^{n+m}$…

Minimal immersion conjecture. The immersion $f$ must be totally geodesic.

Let $\mathbb S^{l}(\frac{1}{2})\times\mathbb S^{l}(\frac{1}{2})$ be the generalised Clifford torus, viewed as a submanifold of $\mathbb S^{2l+2}$, and consider the associated bihar…

Let $M$ be a Riemannian manifold and let a homologically area-minimizing cycle in $M$ have a singular set. White's generic regularity conjecture. Singularities in a homologically a…

Let $M^n$ be a closed minimal submanifold of the unit sphere $\mathbb S^{n+m}$. With respect to a local orthonormal normal frame, let $\mathcal A$ be the fundamental matrix of the…

Let $M^n$ be a closed, minimally immersed submanifold in the unit sphere $\mathbb{S}^{n+m}$ with constant scalar curvature, equivalently with constant length of the second fundamen…

Let $\omega$ be a partial permutation, and let $X_\omega\subset\mathfrak{M}_{m,n}(\mathbb{R})$ denote the corresponding open dense regular part of the real matrix Schubert variety.…

Wulff inequality for minimal submanifolds. The inequality

Let $\Sigma\subset\mathbb{R}^{n+m}$ be an $n$-dimensional smooth minimal submanifold with smooth boundary $\partial\Sigma$. Write $|\Sigma|$ for its $n$-dimensional volume,…

Di Scala's conjecture. The immersion $f$ must be totally geodesic.

Chern's conjecture. The value of $|B|^2$ must lie in a discrete subset of $\mathbb{R}$.

Let $M$ be a compact, simply connected Riemannian manifold whose sectional curvature lies in $(1/4,1]$. A closed minimal submanifold of $M$ is stable when its second variation sati…

Let $M^n$ be an $n$-dimensional compact minimal submanifold of the unit sphere $S^{n+p}(1)$, and let $\overline{\Lambda}_j$ denote the eigenvalues of the closed Laplace–Beltrami pr…

Leung's stronger conjecture. If $n$ is odd and $\sigma(W)<\frac{n+1}{n-1}$, then $W$ is homeomorphic to $S^n$.

Leung's conjecture. If $n$ is odd and $\sigma(W)\leq\frac{n}{n-1}$, then $W$ is homeomorphic to $S^n$.

Let $G$ be a finite set of isometries of $\mathbf{R}^n$. A smooth, closed, $G$-invariant, embedded submanifold of $\mathbf{R}^n$ is one whose boundary-bounded minimal submanifolds…

Leung's weak pinching conjecture. If $n$ is odd and $\sigma(W)\leq\frac{n}{n-1}$, then $W$ is homeomorphic to $S^n$.

Let $B^n$ be the unit ball with boundary $S^{n-1}$. Let $\Sigma$ be a $k$-dimensional minimal submanifold in $B^n$ with boundary $\partial\Sigma\subset S^{n-1}$, and suppose that t…

Let $M$ be an $n$-dimensional minimal submanifold in $S^{n+m}$. Let $\lambda_2$ denote the quantity used in the paper, and suppose that $||\sigma||^2+\lambda_2$ is constant. Pinchi…