99 problems

Let $M$ be a closed minimal hypersurface in a unit sphere, and consider the scalar curvature of $M$. Chern's conjecture. The possible constant values of the scalar curvature of clo…

Let $n\geq 2$, and let $\Sigma^{n}\subset \mathbb{R}^{n+1}$ be a complete minimal hypersurface. Calabi's conjecture. The hypersurface $\Sigma^{n}$ must be unbounded. This is the un…

Let $(M^{n+1},g)$ be a closed Riemannian manifold with $3\leq n+1\leq 7$, and let $\omega_k(M^{n+1},g)$ be its min–max widths. For a generic metric $g$, write a min–max realization…

Let $M^n$ be a closed embedded minimal hypersurface of the sphere $\mathbb{S}^{n+1}$, and let $\lambda_1(M)$ denote the first positive eigenvalue of its Laplacian. Yau's conjecture…

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 $\Sigma^n$ be a smooth closed hypersurface minimally embedded in the unit sphere $\mathbb{S}^{n+1}$ equipped with the round metric, and let $\lambda_1(\Sigma)$ denote the first…

Marques–Neves–Schoen conjecture. There exists $C>0$ such that, for all such hypersurfaces $\Sigma$, one has

Let $(M^{n+1},g)$ be a closed Riemannian manifold with $3\le n+1\le 7$, and let $c>0$. A closed hypersurface has constant mean curvature $c$ if its mean curvature is equal to $c$ a…

A closed Riemannian manifold $(M^{n+1},g)$ is a compact manifold without boundary equipped with a Riemannian metric. Yau's conjecture. Every closed manifold should contain infinite…

A hypertorus is an immersion of ${\mathbf{S}^{1}}\times {\mathbf{S}^{1}}\times {\mathbf{S}^{1}}$ in ${\mathbf{S}^{4}}$. Consider minimal embedded hypertori in ${\mathbf{S}^{4}}$. C…

Let $M^n$ be an embedded closed minimal hypersurface in the unit sphere $S^{n+1}(1)$, and let $\lambda_1(M)$ denote its first Laplace--Beltrami eigenvalue. Yau's conjecture. … Yau'…

Let $M^n$ be a closed, minimally immersed hypersurface of the unit sphere $\mathbb{S}^{n+1}$ with constant scalar curvature. Stronger version of Chern's conjecture. Then $M^n$ is i…

Strong Chern's conjecture. $M^n$ is isoparametric.

Consider the Carlotto–Schulz minimal hypersurfaces with $k=\ell>1$. Their stability index is at least $k^2+6k+8$. Stability-index formula conjecture. The stability index of these h…

Let $(N^{n+1},g)$ be a closed Riemannian manifold with positive Ricci curvature. A closed embedded minimal hypersurface is a smooth hypersurface $M^n\subset N^{n+1}$ with zero mean…

Let $M^{n+1}$ be a closed Riemannian manifold with $3\leq n+1\leq7$, and let $c>0$. A closed hypersurface in $M$ is called a $c$-constant mean curvature ($c$-CMC) hypersurface if i…

Let $M$ be a closed minimal hypersurface in the unit sphere $\mathbb S^{n+1}$ with constant scalar curvature. A minimal hypersurface is isoparametric when its principal curvatures…

In a closed Riemannian 3-manifold, a minimal hypersurface is a smooth, closed, immersed hypersurface with vanishing mean curvature. Yau's conjecture. Every closed Riemannian 3-mani…

Let $(M^3,g)$ be a closed Riemannian $3$-manifold, and let $\{\Sigma_p\}_{p\in\mathbb{N}}$ be the sequence of closed embedded minimal surfaces constructed by the min-max theory, wi…

Let $\mathcal{M}^n\subset \mathbb{S}^{n+1}$ be a compact, embedded minimal hypersurface, and let $\lambda_1(\mathcal{M}^n)$ denote the first nonzero eigenvalue of its Laplace opera…

Let $M^n$ be a compact minimal hypersurface in the unit sphere $S^{n+1}$ with constant scalar curvature, and let $A$ be its shape operator. Consider the possible values of $|A|^2$,…

Let $S^{n+1}$ be the unit $(n+1)$-sphere with its standard round metric, and let $\Sigma^n\subset S^{n+1}$ be a closed embedded minimal hypersurface. Yau's conjecture. The first ei…

Let $M^n$ be an embedded minimal hypersurface in the unit sphere $S^{n+1}$, and let its first eigenfunction and first eigenvalue be understood with respect to the Laplacian on $M$.…

Let $x\colon M^3\longrightarrow\Bbb{S}^4(1)$ be a hypersurface immersion, with normalized mean curvature $H$, second elementary symmetric curvature function $H_2$, and Gauß–Kroneck…

Let $M$ be a Riemannian manifold, and consider area-minimizing hypersurfaces in $M$. Generic smoothness conjecture for area-minimizing hypersurfaces. Area-minimizing hypersurfaces…