33 problems

Let $(M^d,g)$ be a complete asymptotically Euclidean manifold of real dimension $d$ with non-negative scalar curvature and an outermost minimal hypersurface $\Sigma$. Write…

Let $M$ be the closed manifold and let $\Gamma_\infty(M)$ denote the space of smooth Riemannian metrics on $M$. Let $\mathcal{M}_{ei}$ be the set of metrics for which the set of cr…

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…

Bryant's conjecture. $M$ is isoparametric.

Second pinching conjecture. If $S>n$, then $S\geq 2n$.

Let $M^{n+1}$ be as in the equivariant setting, with a compact Lie group $G$ acting by isometries and satisfying … A $G$-homology class is represented by a fixed $G$-hypersurface…

Let $M^{n+1}$ be a closed Riemannian manifold equipped with a compact Lie group $G$ acting by isometries, satisfying … A closed embedded minimal hypersurface is $G$-invariant when…

Let $M$ be a closed three-dimensional manifold. Yau's conjecture. $M$ contains infinitely many immersed minimal surfaces. This conjecture has been resolved: Song proved the asserti…

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…

Let $\Sigma^3\subset B^4\subset\mathbb{R}^4$ be a capillary minimal hypersurface in the unit ball; the free boundary case corresponds to contact angle $\pi/2$. Assume that … for al…

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 $M^{n}$ be a closed immersed minimal hypersurface of the unit sphere $\mathbb{S}^{n+1}$ with constant scalar curvature. A hypersurface is isoparametric if it has constant princ…

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…

Equivariant Yau conjecture. The manifold $M$ contains infinitely many closed embedded minimal $G$-hypersurfaces $\Sigma$ in the given $G$-homology class. This is the equivariant ge…

Let $(M^{n+1},g)$ be a closed Riemannian manifold and let $[\Sigma_0]\in H_n(M^{n+1};\mathbb Z_2)$ be a homology class represented by a closed embedded hypersurface. Homology-class…

Stable Allen–Cahn regularity conjecture. If $n \leq 7$, then the curvatures of the level sets of $u_{\varepsilon_j}$ in these interfacial regions are uniformly bounded along the se…

Let $(M^{n+1},g)$ be the ambient Riemannian manifold under consideration, let $h\in C^{\infty}(M)$, and let $H_{Y_p}$ denote the mean curvature of a hypersurface $Y_p\subset M$. Pr…

Let $n>2$, and let $(S^n,g)$ be an $n$-dimensional Riemannian sphere. A Besicovitch-type geodesic–hypersurface inequality asserts that there is a constant $C(n)>0$, depending only…

Let $x:M^n\rightarrow\mathbb{S}^{n+1}$ be a full, closed, minimal hypersurface in $\mathbb{S}^{n+1}$, with $n\geq 3$. Here $\operatorname{Ind}(M)$ denotes the Morse index of $M$. P…

Schoen–Yau–Schick degree-one dominated $S^2$-stability conjecture. Every Schoen–Yau–Schick manifold $X^n$ with $n\leq 5$ has the degree-one version of dominated $S^2$-stability.

Perdomo's conjecture. One has $\sigma\ge n$. Moreover, $\sigma=n$ if and only if $\left\lvert A\right\rvert^2\equiv n$ and $M$ is a Clifford torus.

Let $N$ be a Riemannian manifold and let $(M(t))_{t>0}$ be a mean curvature flow starting from a generic closed hypersurface. A long-time limit has multiplicity $1$ if it occurs wi…

Let $M^{n+1}$ be a closed smooth manifold, and let $g$ be a Baire $C^\infty$-generic Riemannian metric on $M$. A min-max $g$-minimal hypersurface is a minimal hypersurface generate…

Let $M^{n+1}$ be a closed smooth manifold. A Riemannian metric $g$ is called Baire $C^\infty$-generic if it belongs to a Baire generic subset of the space of smooth Riemannian metr…

Let $M^n$ be a closed immersed minimal hypersurface of the sphere $\mathbb{S}^{n+1}$ with constant scalar curvature. Refined Chern conjecture. Then $M^n$ is isoparametric. The text…