34 problems

Let $(M,J)$ be a complex surface of positive first Chern class polarized by a Kähler class $\Omega$. The positive first Chern class convergence conjecture. There exists an initial…

Let $\varphi$ be a closed $G_2$-structure, meaning a positive closed three-form, on a closed $7$-manifold. Consider the Laplacian flow … A closed $G_2$-structure has sufficiently s…

Let $(\mathcal{X},\omega)$ be a closed Kähler manifold, let $\alpha$ be a closed nonnegative $(1,1)$-form, and let $(\omega(t),\alpha(t))$ solve the coupled flow on its maximal int…

Let $(\mathcal{X},\omega)$ be a closed Kähler manifold and let $\alpha$ be a closed nonnegative $(1,1)$-form satisfying … Let $(\omega(t),\alpha(t))$ solve the Kähler–Ricci flow co…

Let $X$ be a K-semistable Fano manifold, and let $\omega$ denote the initial Kähler form for the Kähler–Ricci flow. The iterated balancing filtration is the filtration referred to…

Let $(X,\omega,\Omega)$ be a compact Calabi–Yau manifold, and let $L_1,\dots,L_r$ be special Lagrangian submanifolds in $X$ with the same phase that intersect transversely. Let…

Donaldson's conjecture. For every such $M$ and $\underline{\omega}$, there exists a family with these properties.

Asymptotic convergence conjecture. The curve $\gamma(\cdot,t)$ converges exponentially fast in the smooth topology to a multiply-covered lemniscate or circle.

Stability classification conjecture. The only dynamically stable solutions are circles for the curve diffusion flow; multiply-covered lemniscates of Bernoulli and multiply-covered…

Global asymptotic-shape conjecture. The rescaled flow $\hat\gamma(\cdot,t)$ converges as $t\to\infty$ exponentially fast in the smooth topology to a limit $\gamma_\infty$. Furtherm…

Let $\mathcal{C}_{\theta}$ be the spherical cap, let $\alpha>\frac{1}{n+2}$ and let $\theta\in(0,\pi)$. Suppose that $h$ is a positive, strictly convex function on…

Let $\mathcal{C}_{\theta}$ be the spherical cap and consider the capillary soliton equation referred to as Eq. … . The preceding theorem shows convergence of the normalized capilla…

Let $M\subset\mathbb S^{n+1}_1$ be a spacelike, compact, star-shaped, and $k$-convex hypersurface. For $1\leq k\leq n$, let $\mathcal A_k$ denote the quermassintegral-type quantity…

Let $\Omega$ be a bounded smooth convex domain in $\mathbb{R}^n$. Let $X(t,x)$ be the flow defined by … where $\nu$ is the unit outer normal, $u(t)$ is the torsion function on the…

Let $m\geq0$ and $N_0\geq2$, where $N_0$ is the rotation number of the initial immersed curve. Consider the area-preserving curvature flow of order $m$. Finite-time blowup conjectu…

Let $m\geq1$. Consider the area-preserving curvature flow of order $m$ for embedded initial curves with length $L_0$ and enclosed area $1$. Immortal-solution conjecture. For every…

Let $G$ be the curvature function defining the flow, and consider convex ancient $G$-flows that are noncollapsing, uniformly two-convex, and compact. Uniqueness conjecture. Up to p…

Let $\mathsf{G}$ be a simple complex Lie group. A left-invariant solution is a solution of the Hermitian curvature flow equation that is invariant under left translations, and…

Let $(M,V,\varepsilon)$ be a marked weighted connected closed surface with $\varepsilon:V\rightarrow\{0,1\}$, and suppose there exists a PL or PH metric generated by a discrete con…

Giga's conjecture. Then $T=\infty$.

Let $M$ be a three-manifold equipped with a negatively curved metric, and let the Cross Curvature Flow (XCF) evolve this metric. A hyperbolic metric is a metric of constant negativ…

Let $X$ be the manifold underlying a Chern-Yamabe flow, and suppose the flow exists on … for some $T>0$. Denote its Chern scalar curvature by $S(x;t)$. Chern-Yamabe flow curvature-…

Let $S$ be a Schrödinger operator evolving with a geometric flow $\mathcal{F}$, and suppose that its eigenvalues are non-decreasing under $\mathcal{F}$. Let $T$ be an operator unit…

Let $S$ be a Schrödinger operator evolving with a geometric flow $\mathcal{F}$. Let $\Delta$ denote the Laplace–Beltrami operator, and suppose that the eigenvalues of $-\Delta$ are…

Suppose $|e|>0$. Let $\eta_1,\eta_2:\mathbb{R}\to\mathbb{R}^2$ be parallel straight lines defining the exterior problem, and let $\gamma_0:(-1,1)\to\mathbb{R}^2$ be an immersed cur…