21 problems

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 $(M^{n+1},g)$ 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 $M$ be an asymptotically flat manifold of low dimension, and let $c$ be a prescribed mean curvature. A closed hypersurface in $M$ has constant mean curvature $c$ when its mean…

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…

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 $M$ be a smooth compact connected embedded hypersurface in $\mathbb R^{n+1}$. The Main Assumption is that whenever $(X',A),(X',B)\in M$ satisfy $A\le B$ and the segment…

Let $(M,g)$ be a spacetime with compact Cauchy surfaces satisfying the strong energy condition. Dilts–Holst conjecture. If $(M,g)$ has a Cauchy surface of constant signed mean curv…

Let $(M,g)$ be a spacetime with compact Cauchy surfaces. CMC existence conjecture. If $(M,g)$ is future timelike geodesically complete and satisfies the strong energy condition, th…

Balmus–Montaldo–Oniciuc constant-mean-curvature conjecture. Any proper biharmonic submanifold in $\mathbb{S}^{m+1}$ is CMC.

Let $\gamma:S^n\to {\mathbb R}_{>0}$ be a $C^2$ function, and let $X:M\to {\mathbb R}^{n+1}$ be a closed CAMC hypersurface. For each $r=1,\ldots,n$, assume that the $r$-th anisotro…

Let $M$ be a closed hypersurface with constant mean curvature and constant scalar curvature in the unit sphere $\mathbb{S}^{n + 1}$. Generalized Chern conjecture. For each $n$ and…

Let $(M,g,K)$ be an initial data set, equipped with the $C^{2,\alpha}$ norm, and let its associated spacetime be the evolution determined by the data. Open-set conjecture. There is…

Let $\mathcal{V}$ be a cosmological spacetime. A slice has constant signed mean curvature if its mean curvature is constant with a fixed sign convention, while a CMC slice is a sli…

Let $\mathcal{V}$ be a cosmological spacetime. A timelike path is incomplete when it has finite length. Timelike incompleteness criterion. If every future or every past timelike pa…

Let $\mathcal{V}$ be a globally hyperbolic, cosmological spacetime. A future ray or past ray is a complete timelike path in the corresponding time direction, and a line is a comple…

Tilted-unduloid conjecture. For every $\alpha\in I\setminus\{\alpha_0\}$, there exists $\Sigma^\alpha\in\mathcal{M}(h_1,h_2^\alpha)$ such that $\widetilde{\Sigma^\alpha}$ extends t…

Malec–O Murchadha conjecture. Given any constant mean curvature $H$, the Kruskal extension can be foliated by the TSS-CMC hypersurfaces $\{\Sigma_H\}$. The paper presents this as t…

Let $M$ be a closed hypersurface with constant mean curvature $H$ in $\mathbb{S}^{4}$, and let $S$ denote the squared length of its second fundamental form. Write $\beta(3,H)$ for…

CMC index conjecture. Either