80 problems

Let $(M^n,g)$ be a compact Kähler manifold of complex dimension $n$, let $comega_g$ be a Kähler metric, and let $comega_{cvarphi_s(t)}$ be a family of Kähler metrics in the Kähler…

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 $M_t=X(\mathbb{S}^n,t)\subset\mathbb{R}^{n+1}$ be a family of smooth, strictly convex, closed hypersurfaces evolving by Gauss curvature flow, with $K$ the Gauss curvature and…

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…

De Giorgi's conjecture. Any initial $n$-dimensional smooth submanifold $\varphi:M\to\mathbb{R}^{n+m}$ evolves by the gradient flow of $\mathcal{G}_k$ without developing singulariti…

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…

A translator is a translating solution of Gauss curvature flow, represented as a graph over a domain $\Omega\subset \mathbb{R}^2$. The boundary $\partial\Omega$ is the boundary of…

Folklore conjecture. Type-I behavior is generic for compact embedded solutions.

De Giorgi's nonsingularity conjecture. The evolving hypersurface does not develop singularities during the flow.

Suppose $|e|>0$, and let $\gamma:(-1,1)\times[0,T)\to\mathbb{R}^2$ be a solution to the vanilla elastic flow (E). The relevant theorem asserts global existence and exponential conv…

Let $M$ be a compact Kähler manifold, let $[\omega]$ be a Kähler class on $M$, and consider a global $C^{1,1}$ minimizer of the $K$-energy in $[\omega]$. Chen's conjecture. Every s…

Odd-dimensional all-time existence conjecture. The solution to the Ricci Yang-Mills flow exists for all time.

The $\chi$-flow conjecture. The $\chi$-flow on $\Sigma\subset Q$ takes $\Sigma$ diffeomorphically onto the zero section.

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…

Embeddability preservation conjecture. Embeddability is preserved under the Cartan flow without any conditions.

Donald's conjecture. If this necessary condition is satisfied, then there exists a critical point for $J_{\chi}$ in that Kähler class. This is an existence assertion for solutions…

Streets–Tian conjecture. In analogy with the Kähler–Ricci flow, the solution of the pluriclosed flow with initial metric $g_0$ exists on the maximal interval $[0,\tau(g_0))$.

Let a compact group act in a finite-dimensional setting with moment map, and let the flow be the gradient flow of a convex function of that moment map. Conjecture on convex moment-…

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…

Given an artinian abelian category $\mathcal A$ and a homomorphism $X \colon K_0(\mathcal A) \to \mathbb R$ positive on the class of every non-zero object, let the iterated HKKP fi…

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…