174 problems

Let $\Sigma\subset\mathbb S^3$ be the minimal Möbius band specified in White's conjecture, and let $C(\Sigma)=\{tx:t\geq 0,\ x\in\Sigma\}\subset\mathbb R^4$. The conjecture asserts…

Let $C\subset\mathbb{S}^3$ be a great circle. Is every embedded non-orientable minimal surface $\Sigma\subset\mathbb{S}^3$ with boundary $\partial\Sigma=C$ congruent to the Lawson…

Every smooth properly embedded minimal surface $\Sigma\subset \overline{\mathbb{B}}^{3}$ that is topologically an annulus, satisfies $\partial\Sigma\subset \mathbb{S}^{2}$, and mee…

For each $n\ge 4$ and each $\sigma>0$, there should exist a constant $C(n,\sigma)<\infty$ such that every smooth closed $n$-manifold with isotropic curvature bounded below by…

Determine whether, for every dimension $n$ and every algebraic projection multiplicity $Q$, there exists a constant $C(n)$ such that every confined area-minimizing rectifiable curr…

Let $M$ be a hyperbolic open Riemann surface, meaning an open Riemann surface that is not parabolic. A proper harmonic map is a harmonic map whose inverse image of every compact se…

Let $u$ be a bounded solution of the Allen–Cahn equation … Assume that $u$ is monotone in one direction, namely $u_{x_n}>0$. De Giorgi's conjecture. If $n\leq 8$, then $u$ is one-d…

Let $B^3\subset\mathbb{R}^3$ be the unit ball, and let $\Sigma_{\mathrm{cc}}$ denote the critical catenoid, the free boundary minimal annulus obtained by scaling the Euclidean cate…

Let $M$) be a closed surface minimally immersed in $\mathbb{S}^N$ such that its image is not contained in any hyperplane of $\mathbb{R}^{N+1}$. Set $S=|h|^2$, where $h$ is the seco…

Low-conformal-volume isotopy conjecture. If

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 $T\subset S^3$ be an embedded minimal torus, and let $T_{\mathrm{Cl}}$ denote the Clifford torus in $S^3$. Lawson's conjecture. There exists an isometry $F$ of $S^3$ such that…

Let $M^{n+1}$ be a closed manifold with a generic Riemannian metric, where $3\le n+1\le 7$. Consider closed minimal hypersurfaces obtained by min-max methods, and call a component…

A critical surface is a Heegaard surface with the criticality property defined in the paper, and an index 2 minimal surface is a minimal surface whose second variation has index 2.…

Let $M$ be a manifold diffeomorphic to $S^3$, equipped with a Riemannian metric. An embedded minimal two-sphere in $M$ is an embedded minimal submanifold diffeomorphic to $S^2$. S.…

Multiplicity-one conjecture. The convergence to the unstable minimal surface must have multiplicity $1$.

Let a strongly irreducible Heegaard surface lie in a Riemannian three-manifold. Pitts–Rubinstein conjecture. It is either isotopic to a minimal surface of index at most $1$, or iso…

Labourie's conjecture. Does there exist a unique $\rho$-equivariant minimal surface in $G/K$?

Let $M\subset\mathbb{R}^3$ be a connected, complete embedded minimal surface of finite genus with compact, possibly empty, boundary. Embedded Calabi–Yau conjecture for finite genus…

Rubinstein's conjecture. $H$ is either isotopic to a minimal surface or isotopic to the oriented double cover of a non-orientable minimal surface with a vertical handle attached.

Let $M$ be a complete surface of finite genus embedded in $\mathbb{R}^3$ with constant mean curvature. Say that $M$ has at most cubical area growth if there is a constant $C$ depen…

Let $\mathcal W(\Sigma)=\int_{\Sigma}(1+H^2)\\,d\mu$ denote the Willmore energy of a compact surface $\Sigma\subset S^3$, where $H$ is its mean curvature. Willmore's conjecture. Ev…

An open surface is a non-compact surface without boundary. Let $S$ be an open, connected, orientable surface. Ros's conjecture. Every such surface can be properly and minimally emb…

Let $V$ be a stationary integer rectifiable varifold, let $x_0$ be a point with a unique flat tangent cone, let $\Theta(V,x_0)$ denote its density at $x_0$, and let…

Let $(M,h_0)$ be a closed hyperbolic manifold of dimension $n\geq 3$, and let $N$ be a closed $n$-manifold with Riemannian metric $g$ whose scalar curvature satisfies…