48 problems

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…

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…

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

Let $X:M^n\to\mathbb{R}^{n+1}$ be a complete self-shrinker, and let $S$ be the squared norm of its second fundamental form. Cheng–Wei–Yano's conjecture. If $S$ is constant, then…

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 $M$ be a complete self-shrinker in $\mathbb{R}^n$, and let the drift Laplacian be … The coordinate functions satisfy $\mathcal{L}x_i=-\frac{1}{2}x_i$, so they are eigenfunction…

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…

Rotational self-shrinker uniqueness conjecture. There are no embedded rotationally invariant self-shrinkers of topological type $\mathbb{S}^1 \times \mathbb{S}^{n-1}$ in…

Berchenko-Kogan's exponential entropy-limit conjecture. The entropy values of the $2$-dimensional Angenent torus appear to lie on an exponential curve converging to $1.8512167$.

Spherical self-shrinker conjecture. The round sphere should be the only embedded self-shrinker that is diffeomorphic to a sphere.

A self-shrinker is a surface in $\mathbb R^3$ that evolves under mean curvature flow by homothetic shrinking. Its index is the number of negative directions of the associated shrin…

A self-shrinker is a surface in $\mathbb R^3$ evolving under mean curvature flow by homothetic shrinking; its Gaussian area is the weighted area functional whose critical points ar…

Let $n\leq 6$ and let $\Sigma\subset\mathbb{R}^{n+1}$ be a smoothly embedded self-shrinker whose ends satisfy the nice ends hypothesis: each end is either smoothly asymptotic with…

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…

Spherical self-shrinker conjecture. The round sphere should be the only embedded self-shrinker which is diffeomorphic to a sphere.

Conjecture. $X$ is isometric to one of

Let $P$ be any regular polyhedron, and form its doubling by joining two copies of $P$ with necks at the center of each face. Chopp's polyhedral doubling conjecture. The resulting d…

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…

Let $\Theta_g\subset\mathbb{R}^3$ be the complete, embedded, noncompact self-shrinker of genus $g$ invariant under the dihedral group $\mathbb{D}_{g+1}$ constructed in the paper. A…

Colding–Minicozzi's conjecture. The only such self-shrinkers are the round cylinders $S^k(\sqrt{2k})\times\mathbb{R}^{n-k}$. If true, this would imply, together with the codimensio…

Generalized-cylinder conjecture. The only self-shrinker with asymptotically cylindrical ends is the generalized cylinder.

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 a closed immersed self-shrinker in the plane be an Abresch–Langer curve. Abresch–Langer conjecture. After an outward perturbation, the curve becomes round under mean curvature…