201 problems

A Calabi–Yau manifold is a Kähler manifold equipped with a nowhere-vanishing holomorphic volume form, and a Lagrangian submanifold is stable in an appropriate sense as specified by…

Let the space of embedded two-spheres in $\mathbb{R}^3$ be the space of embeddings of $S^2$ into $\mathbb{R}^3$, with its natural topology. Smale's conjecture. In one of its many f…

Consider the mean curvature flow of embedded hypersurfaces, and let a blowup limit mean a limit obtained by parabolic rescaling near a singularity, counted with its varifold multip…

Let $S_t\subset\mathbb{R}^3$ be a family of compact surfaces evolving by mean curvature flow, with generic initial data $S_0$. A singularity is spherical or cylindrical when its ta…

Let $M_t$ be a closed smooth embedded mean curvature flow in $\mathbb{R}^{n+1}$, with $n dleq 6$, and suppose the flow develops a first finite singular time. Mean-curvature blow-up…

For a hypersurface $\Sigma$ in Euclidean space $\mathbb{R}^{n+1}$, define its entropy by … Here the supremum is over all $t_0>0$ and $x_0\in\mathbb{R}^{n+1}$. Entropy-minimizing sp…

Let $\Sigma^n\subset\mathbb{R}^{n+1}$ be a complete properly embedded self-shrinker satisfying the self-shrinker equation, and let $\lambda_\rho(\Sigma)$ denote its weighted first…

Let $\Sigma$ be an asymptotically cylindrical shrinker in $\mathbb{R}^{n+1}$. Ilmanen's conjecture. The shrinker $\Sigma$ must coincide with the standard cylinder. The source prese…

Ecker's monotonicity conjecture. In the case of mean curvature flow for compact embedded hypersurfaces $M_t$ in $\mathbb{R}^{n+1}$, the entropy satisfies

White's conjecture. The only convex eternal solutions are the translators.

Genus Reduction Conjecture. The genus of $M(t)$ strictly decreases at the moment of a singularity, unless the singularity is a neckpinch or shrinking sphere.

Mean curvature flow is an evolution of hypersurfaces by their mean curvature vector; its singular set records the spacetime points where the flow develops singularities. In the mea…

Consider a cylindrical singularity of mean curvature flow. A neighborhood is mean-convex when the flow in that neighborhood has the corresponding one-sided mean-convex orientation.…

Consider the outer or inner flow of an embedded surface through singularities. A singularity is an isolated singularity if it is isolated in the relevant spacetime singular set. Is…

Let $\Sigma\subset \mathbb R^N$ be a closed $n$-dimensional self-shrinker. The Colding-Minicozzi entropy should satisfy … This conjecture concerns the entropy-minimizing closed sel…

Let a cylindrical singularity be a singularity of mean curvature flow modeled on a cylinder. Mean convex neighborhood conjecture. Every cylindrical singularity has a mean convex ne…

Let a singular mean curvature flow have a Type II blowup, and let its blowdown be the rescaled asymptotic model of that Type II blowup. A Type I blowup is a blowup obtained at the…

An $n$-dimensional smooth hypersurface $\Sigma^n\subseteq\mathbb R^{n+1}$ is a self-shrinker if … for all $x\in\Sigma$, where $H_\Sigma$ is the mean curvature with respect to the o…

Let an ancient oval be an ancient compact noncollapsed mean curvature flow that is not self-similar, and let $1\leq k\leq n$. An ancient oval is…

Let $X:M\to \mathbb{R}^{n+1}$ be an $n$-dimensional complete self-shrinker in $\mathbb{R}^{n+1}$. Suppose that the scalar curvature of $M$ is constant and positive. Classification…

Folklore conjecture. Type-I behavior is generic for compact embedded solutions.

For $N\geq 2$, let $\Omega_0$ be a bounded open subset of $\mathbb{R}^N$ with boundary $\Gamma_0$. Let $u$ solve the heat equation with initial data $\mathbf{1}_{\Omega_0}$, and de…

Sharp constant conjecture. The sharp value of the universal constant $C_1$ in the Colding–Minicozzi codimension bound is

Let $M_{0}$ be an $n$-dimensional complete submanifold in the sphere $\mathbb{S}^{n+q}\left(1/\sqrt{c}\right)$. Let $h$ be its second fundamental form, $H$ its mean curvature vecto…

Finiteness conjecture. The space-time singular set has only finitely many components.