11 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…

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.

Donald's conjecture. The distance satisfies

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

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 $M$ 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…

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

Calabi's flow conjecture. The Calabi flow exists for all time and converges to a csc metric or, in an appropriate sense, to an extremal metric.

Let $X$ be a toric surface with Delzant polygon $P$, and let $u(t)$ be a solution of the Calabi flow equation on $P$, defined on its maximal interval of existence $[0,T)$. Global e…

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 $c_1(L)$.