169 problems

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…

Let $\mathbb{B}$ be the unit ball and let $\Delta\subset\mathbb{B}$ be the properly embedded one-manifold given in the proof of Theorem. Let $\overline{B}$ be a smooth compact Riem…

Genus-zero conjecture. $M$ is a Riemann minimal example.

The continuity-based criterion. Under these assumptions, $A$ does not have property (iv). The conjecture proposes that the conclusion of Theorem 2 remains valid when the relevant g…

Storm's local-minimality conjecture. If $M$ is locally minimal among such manifolds, then its boundary is totally geodesic.

Let $B$ be a closed geodesic ball in a Riemannian three-manifold, and let $M$ be a minimal surface with boundary $\partial M \subset \partial B$. Suppose that $M$ is properly embedde…

Let $M$ be a connected properly immersed minimal surface in ${\mathbb R}^3$, and let $A(M)$ denote its area growth constant. Let $S_\theta$, for $\theta \in (0,\frac{\pi}{2}]$, den…

Let $M$ be a properly immersed minimal surface in ${\mathbb R}^3$ with quadratic area growth, meaning that its area growth constant $A(M)$ is finite. A limit tangent cone at infini…

Direction-of-approach conjecture. The surface parts of the limit image should depend only on the boundary point of moduli space approached, whereas the complex of straight line seg…

Uniqueness conjecture. It may be possible that there is only one boundary point of the compactified moduli space having a finite-energy limit of map energies for the given set of m…

Monotonicity conjecture. The angle $\alpha(p)$ is a monotone function for

Meeks–Sullivan conjecture. The surface $M$ is parabolic.

Delaunay-chain transition conjecture. The first phase transition as small equal, or close to equal, volumes grow is from the standard double bubble to a chain of two bubbles bounde…

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 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…

Let $\Sigma$ be the singular set of the anisotropic obstacle problem, and let $A$ be the coefficient matrix with ellipticity constants. $C^{1,\alpha}$-rectifiability conjecture. Th…

Fujimoto conjecture. These polynomials are in general position: any $m$ polynomials chosen from them are linearly independent over $\mathbb{C}$. This conjecture is the algebraic in…

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.…

Let $\Sigma_a$ be the critical spherical catenoid with parameter $a>1/2$. Decompose its Robin Morse index into Fourier modes $|k|=0$, $|k|=1$, and $|k|\geq 2$. Mode-by-mode index c…

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 $B^3\subset\mathbb{R}^3$ 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 $T\subset S^3$ 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 $u$ 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…

Del Pino–Kowalczyk–Wei conjecture. Outside a large ball, each level set of $u$ has finitely many components, each asymptotic either to a plane or to a catenoid; after a rotation of…