86 problems

For a suitable vector-valued coupled Allen–Cahn evolution with diffuse-interface parameter $\varepsilon\to0$, prove that the associated diffuse interfaces converge globally and unc…

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 $M_t$ 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…

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

Consider mean curvature flow with generic initial data, meaning initial data outside an appropriate exceptional subset. A spherical singularity has spherical tangent model, and a c…

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…

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

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 $M_{0}$ 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.

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…

Let $L$ be a graded Lagrangian submanifold of a Calabi–Yau $n$-fold $X$, with the phase of the holomorphic volume form chosen so that the cohomological phase satisfies $\phi(L)=0$.…

Generalized Liu–Terng conjecture. (i) If $A(t)$ and $H(t)$ satisfy the inequality referred to as for some $\epsilon\in(0,1/(2\kappa))$ instead of $\epsilon_F$, then $M$ is an equif…

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

Let $(M,\omega,\Phi)$ be a Calabi–Yau $n$-fold, and let $\eta$ be an $n$-dimensional homology class in $M$. For an oriented exact Lagrangian $L$ in $M$, write … where…

Generic nondegeneracy conjecture. For a generic compact embedded initial hypersurface $L_0$, the mean curvature flow $L_t$ admits only spherical and nondegenerate cylindrical singu…

Let $M_t\subset\mathbb{R}^{n+1}$ be an ancient oval whose tangent flow at $-\infty$ is $\mathbb{R}^{k}\times S^{n-k}$. A $k$-oval is an ancient oval with this tangent flow whose hi…

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 a mean curvature flow of surfaces approaching its first singular time, and let intrinsic diameter denote the diameter measured in the evolving surface's intrinsic metric.…

A mean curvature flow develops tangent flows by rescaling near a singularity. A sequence of rescaling factors is a sequence $\lambda_i\to\infty$ used to obtain a blowup limit. Uniqu…

Let $\mathcal{M}$ be a rescaled mean curvature flow in $\mathbb{R}^{n+1}$ without boundary. The source distinguishes asymptotic cases for such flows, including a case of super-expo…

Let $n\geq 3$ and $k\geq 2$. For a mean curvature flow $\mathbf{M}$ in $\mathbb{R}^{n+1}$, let $\mathcal{S}_{k}(\mathbf{M})_+$ denote its positive part of the $k$-th cylindrical si…

Let $M_t$ be any mean curvature flow in $\mathbb{R}^3$. A genus reduction conjecture asserts that the genus of $M_t$ strictly decreases at the time of a singularity, unless the sin…

Let $M^i\subset\mathbb{R}^3$, $i=1,2$, be two smooth, connected, compact hypersurfaces in $\mathbb{R}^3$. Denote by $M^i_t$ the outermost mean curvature flows emanating from $M^i$.…