23 problems

Sublinear multiplicity conjecture. There exist $\alpha<1$ and a constant $C_{\alpha}=C_{\alpha}(\alpha,\gamma)$ such that

Let $M^n\subset\mathbb{R}^{n+1}$ be a connected, complete, embedded self-shrinker with polynomial volume growth, and let $A$ be its second fundamental form. Sharp Gaussian curvatur…

Let $\Sigma\subset \mathbb R^3$ be a closed self-shrinker of positive genus, and let $\Sigma_C\subset \mathbb S^3$ be the Clifford torus. One should have … This conjecture is motiv…

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 $X:M^n\rightarrow\mathbb{R}^{n+1}$ be an $n$-dimensional complete self-shrinker, meaning that its mean curvature vector satisfies … Suppose that the squared norm of its second…

For sufficiently large $n\in\mathbb{N}$, consider a complete, embedded, noncompact self-shrinker $\Phi_g\subset\mathbb{R}^3$ for mean curvature flow, where $g=2n-1$ denotes its gen…

A self-shrinker is a complete embedded shrinker for mean curvature flow. The no-cylinder conjecture. The only complete embedded shrinker with a cylindrical end is the round cylinde…

Let $\Sigma$ be an embedded shrinking soliton in $\mathbb{R}^3$, and suppose that $\Sigma$ is not a round cylinder. No cylinder conjecture. Then $\Sigma$ cannot have an end asympto…

Let $\Sigma$ be a compact, embedded, nonround self-shrinker in $\mathbb{R}^{n+1}$. An associated-flow conjecture. There exists an associated nonsoliton ancient flow … This proposes…

Rigidity conjecture. Then $u^\alpha(x)$ is a linear function for each $\alpha=1,\ldots,n$.

Low-entropy cylinder conjecture. There exists $\epsilon>0$ such that, for $n\leq 4$ and in any codimension, the only shrinkers with entropy $<\lambda({\boldsymbol S}^1)+\epsilon$ a…

Lowest-entropy conjecture for generalized cylinders. For any codimension and $n\leq 4$, the round generalized cylinders $\mathbb S^k_{\sqrt{2k}}\times\mathbb R^{n-k}$ are the shrin…

Plane-or-cylinder asymptotic conjecture. As $\lambda\to\infty$, $\Sigma-\lambda\mathbf{v}_0$ converges locally smoothly to a plane or a self-shrinking cylinder of multiplicity one.

Asymptotic structure conjecture. For each $U_i$, either, as $\lambda\to\infty$, $\lambda^{-1}U_i$ converges locally smoothly to a cone smooth except at the origin $\mathbf{0}$, or…

Let … denote the energy, and let … -self-shrinker solution. Monotonicity conjecture. When $>0$, … . This extends the monotonicity known for the self-shrinker case $=0$, attributed…

Let $M^n\subset\mathbb{R}^{n+1}$ be a closed hypersurface with $n\leq 6$, and let $\lambda(M)$ denote its entropy. Let $\lambda(\mathbb{S}^n)$ be the entropy of the round sphere. E…

Let $\Sigma^n\subset\mathbb{R}^{n+1}$ be a smooth, complete, embedded self-shrinker, meaning that its mean curvature satisfies … Assume that $\Sigma$ has polynomial volume growth a…

Let $\Sigma=\mathbb{R}^n\subset\mathbb{R}^{n+1}$ be a self-shrinker with $n\geq 3$. Let $r_1$ be the dimension-dependent radius from the cited remark. Then there should exist a num…

Let $\Sigma^n\subset\mathbb{R}^{n+1}$ be a smooth embedded self-shrinker with entropy at most $\alpha$, where $\alpha>1$. Let $B_R$ be the Euclidean ball of radius $R$, and let $H$…

Let $\Sigma^n\subset\mathbb{R}^{n+1}$ be a self-shrinker, and call it non-flat if it is not a hyperplane. Let $\lambda$ denote entropy. Colding–Ilmanen–Minicozzi–White low-entropy…

Let $M^n\subset \mathbb{R}^{n+1}$ be a closed hypersurface, and let $\lambda$ denote entropy. The cited theorem asserts that, for a closed self-shrinker, the round sphere minimizes…

Ilman's cylindrical rigidity conjecture. If one end of $\Sigma$ is asymptotic to a cylinder, then $\Sigma$ is isometric to the self-shrinking cylinder.

Ilman's asymptotic-end conjecture. There exists $R>0$ such that $\Sigma\setminus B_R$ decomposes into a finite number of ends $U_j$, and for each $j$ either: