51 problems

Let $(S^2,g)$ be a Riemannian $2$-sphere, and let $kappa>0$. A closed curve has constant geodesic curvature $kappa$ if its geodesic curvature is equal to $kappa$ at every point. Ar…

Let $\mathbb S^3$ be the 3-sphere, and let a rotational torus mean a torus invariant under a rotational symmetry of $\mathbb S^3$. Pinkall–Sterling conjecture. The rotational torus…

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 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 $M^{n+1}$ 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…

Homological triviality conjecture. For $j$ sufficiently large, $S_j$ is homologically trivial.

Let $\Pi\subset\mathbb{R}^{n+1}$ be a hyperplane and let $\Sigma\subset\Pi$ be an $(n-1)$-sphere. Consider compact embedded hypersurfaces in $\mathbb{R}^{n+1}$ with constant mean c…

Let $S$ be a compact constant-mean-curvature surface in Euclidean space $\mathbb{R}^3$ whose boundary is a circle. The surface is assumed either to have genus zero or to be embedde…

Unduloid-extremizer conjecture. In the case $x=0$ and $y<1$, the maximizers and minimizers correspond to the unduloid immersions.

Cylinder-extremizer conjecture. These solutions are exactly those corresponding to the maximizers and minimizers of $\mathcal{F}_{q_D}$, when all spinors are normalized to have…

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 $V$ be a $2+1$-dimensional maximal globally hyperbolic flat spacetime with compact Cauchy surface of genus at least $2$, and let $M_\tau$ be the foliation of $V$ by constant me…

Let such a cosmological spacetime have compact Cauchy surfaces. A constant mean curvature (CMC) Cauchy surface is a Cauchy surface whose mean curvature is constant, and a foliation…

Let the sub-Lorentzian Heisenberg group be equipped with its horizontal distribution and sub-Lorentzian metric. Consider the family of smooth, acausal, boost-symmetric surfaces wit…

Let $(M^{n+1},g)$ be a closed Riemannian manifold with $3\le n+1\le 7$. A closed hypersurface encloses half the volume of $M$ when one of the regions it bounds has volume equal to…

Let $S^1\times S^1\times S^1$ be the three-torus, and let the hypertorus denote the standard embedding of this product in the four-sphere $\mathbf{S}^{4}$. Carlotto–Schulz conjectu…

Let $\mathcal{N}$ be an asymptotically Schwarzschildean lightcone. An asymptotic foliation by strictly stable STCMC surfaces is a foliation of the asymptotic region of…

Let $\mathcal{N}$ be an asymptotically Schwarzschildean lightcone. Let $\Sigma$ be a strictly stable STCMC surface with area radius $\rho$, and let $B_\sigma(B_1,B_2,B_3)$ denote t…

Nonexistence conjecture. There is no such hypersurface satisfying

Let $(M^{n+1},g)$ be a closed Riemannian manifold with $3\leq n+1\leq 7$, and let $v\in(0,\operatorname{Vol}(M))$. Twin-volume conjecture. The manifold $M$ admits at least two dist…

Let $p,q\geq2$, and let a global solution curve $\gamma(t)=(x(t),y(t))$ of … where $W(t)=\sqrt{(x'(t))^2+(y'(t))^2}$, generate an $O(p)\times O(q)$-invariant constant mean curvatur…

Let $(M^{n+1},g)$ be a closed Riemannian manifold. A constant mean curvature surface encloses half the volume when it is the boundary of a region of volume…

Let $(M^{n+1},g)$ be a closed Riemannian manifold, and let $\tilde c(p)$ denote its phase-transition half-volume spectrum. A metric is generic if it belongs to the generic class in…