85 problems

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 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 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…

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…

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 $M^i$.…

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…

Let an embedded hypersurface evolve by mean curvature flow in $\mathbb{R}^{n+1}$, with $n dleq 6$, and consider blow-ups at singularities. Ilmanen's smooth-blowup conjecture. T…

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

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 $\gamma(t)$ be a gradient flow line for an analytic function, and suppose that it has a limit point. Arnold--Thom gradient conjecture. The limit … exists. This is a stronger, fi…

Let $N$ be a Riemannian manifold and let $(M(t))_{t>0}$ be a mean curvature flow starting from a generic closed hypersurface. A long-time limit has multiplicity $1$ if it occurs wi…

Standard-flow simplex conjecture. The invariant set

Let $s>0$ be fixed, and let $M_i$ be an annuloid in the family $\mathcal A$ from Corollary, with necksize $s$ and inner width $b_i\to\infty$. After suitable vertical translations, c…