40 problems

A compact, simply connected Riemannian manifold $M$ is $1/4$-pinched when its sectional curvatures satisfy the quarter-pinching condition. A minimal submanifold is stable when the…

Yau's conjecture. The volume of one of the minimal Clifford tori $M_{k,n-k}$ gives the lowest volume among all non-totally geodesic closed minimal hypersurfaces of…

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…

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

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

Let $M^{n+1}$ be a Riemannian manifold and let $\Gamma$ be a closed $\ell$-dimensional submanifold. Suppose there are intervals $I_k=(\rho_k',\rho_k”)$ tending to $0$, and for ever…

Let $M^{n+1}$ be a Riemannian manifold and let $\Gamma$ be a closed $\ell$-dimensional submanifold, with $1\leq \ell\leq n$. Suppose there is a sequence of hypersurfaces $\Sigma_j$…

Let $M^{n+1}$ be a Riemannian manifold and let $\Gamma$ be a closed embedded $\ell$-dimensional submanifold. Suppose there are sequences of intervals $I_k=(\rho_k',\rho_k”)$ tendin…

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…

Lu's second-gap conjecture. There is a constant $\varepsilon(n,q)>0$ such that

Let $M^n$ be a closed immersed minimal submanifold of the unit sphere $\mathbb{S}^{n+q}$. Let $S_\alpha$ be the shape operators with respect to an orthonormal basis…

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.…

A minimal submanifold is a submanifold whose mean curvature vector vanishes. The codimension-two isoperimetric conjecture asserts that the classical isoperimetric inequality in Euc…

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 $\mathbb{R}$.

Ohnita's conjecture. There exists no stable rectifiable $p$-current in $M$ for

Sharp isoperimetric inequality conjecture. The sharp isoperimetric inequality holds for minimal submanifolds in $\mathbb{R}^{n+1}$.

Let $M$ be a homological sphere with positive sectional curvature. A closed minimal submanifold of $M$ is stable if the second derivative of its volume is nonnegative under every v…

Let $\Omega$ be an $n$-dimensional minimal submanifold of $\mathbb{R}^N$, with boundary $\partial\Omega$, and let $B^n$ denote the unit ball in $\mathbb{R}^n$. Minimal-submanifold…

Let $M^3$ be a minimal hypersurface in $\mathbb{S}^4$ with constant norm of the second fundamental form, without assuming that $M$ is compact. Bryant's conjecture. The hypersurface…

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…

The hyperbolic Ginzburg–Landau equation is considered in the regime where the renormalized measures associated with solutions constructed by an approximating scheme are expected to…