10 problems

Let $M$ be a topological $2$-sphere, and let $g_i=e^{2u_i}g_0$ be a family of conformal metrics with bounded area and energy. Assume … Suppose there is exactly one bubble point…

Calabi–Chen conjecture. Initiating from any smooth Kähler potential, the Calabi flow always exists globally.

Chen's short-time regularization conjecture. Given any initial $C^{1,1}$ Kähler potential, the Calabi flow admits a short-time solution and it immediately becomes smooth when $t>0$…

Donaldson's conjecture. If the Calabi flow exists for all time and there exists a cscK metric in the Kähler class, then the Calabi flow converges to a cscK metric.

Chen's conjecture. The Calabi flow exists for all time.

Let $(X,\omega)$ be a compact Kähler manifold, let $\mathcal H_\omega$ denote the space of Kähler potentials in the class $[\omega]$, and let…

Let $(M,\omega,\chi,J)$ be the initial twisted Kähler data, and suppose that the twisted Calabi flows have global existence. The transformed data along the flow may be written…

Let $\varphi$ be a smooth initial Kähler potential, and let the twisted Calabi flow be the evolution equation introduced in the paper for the corresponding Kähler metrics and twist…

Let $X$ be a Kähler manifold with Kähler class $[\omega]$. Suppose that there exists an extremal metric $\omega_0\in[\omega]$. Let $\omega_1\in[\omega]$ be a Kähler metric invarian…

Toric modified-Calabi-flow conjecture. The modified Calabi flow converges exponentially fast to an extremal metric in $c_1(L)$.