3. Geometric Flows
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Flow Convergence from Perturbation of Elliptic Solution
Problem 3.1.
[Albert Chau] Suppose $(M,\omega)$ is a complete Kaher manifold with bounded geometry, and $u\in C^{\infty}(M)$ solves \[ (\omega+\sqrt{-1}\partial\overline{\partial}u)^{n}=e^{f}\omega^{n}, \] and satisfies \[ C^{-1}\omega\leq\omega+\sqrt{-1}\partial\overline{\partial}u\leq C\omega \] \[ |f|(z)\leq\frac{C}{d_{\omega}^{s}(z_{0},z)} \] as $d_{\omega}(z_{0},z)\to\infty$, for some fixed point $z_{0}\in M$. Then does the flow \[ \partial_{t}u=\log\left(\frac{(\omega+\sqrt{-1}\partial\overline{\partial}u)^{n}}{\omega^{n}}\right)-f \] have long time existence? Does it converge to a solution of the corresponding elliptic equation?
This is known when $s>2$. It is also known by Li-Tam that a longtime solution exists assuming $(M,\omega_{0})$ is a complete Kahler manifold with nonnegative holomorphic bisectional curvature and maximal volume growth. Moreover, in this case, we have \[ |Rm|_{\omega_{t}}\leq\frac{C}{t} \] for all $t>0$. However, it may not be known whether $\omega_{t}\to\omega$ as $t\to\infty$.
There is always a solution of $(\omega+\sqrt{-1}\partial\overline{\partial}u)^{n}=e^{f}\omega^{n}$ if $s\in(0,2)$ and $(M,\omega)$ is asymptotically conical. -
Kahler-Ricci Tangent Flows
Problem 3.2.
[Max Hallgren] Given a closed Kahler-Ricci flow $(M,(\omega_{t})_{t\in[0,T)})$, under what conditions are tangent flows complex analytic spaces? It is known if $M$ is Fano or if $n=2$. It is conjectured to be true starting with any projective Kahler-Ricci flow.
What about given a limit of polarized Kahler-Ricci flows? With extra assumptions, this was shown by Chen-Wang. -
Maximal Existence Time for Pluriclosed Flow
Problem 3.3.
[Anna Fino] A pluriclosed metric is a positive $(1,1)$-form $\eta$ such that $\sqrt{-1}\partial\overline{\partial}\eta=0$. The pluriclosed flow is \[ \partial_{t}\eta=(Rc_{B}(\eta_{t}))^{1,1}, \] where $Rc_{B}$ is the Ricci curvature of the Bismut connection. Short time existence is known, and there is an upper bound on the first singular time $T$ in terms of cohomological data (the Appeli cone). Is this upper bound actually equal to the first singular time? This is known assuming some symmetry.
Do there exist static solutions on any compact complex manifold with $n>3$? It is known when $n=2$ and $n=3$ (very recent). -
Expanding Solitons from Divisorial Contractions
Problem 3.4.
[Max Hallgren] Consider a Kahler-Ricci Flow $(M^{2},(g_{t})_{t\in[0,T)})$ developing a finite-time singularity with $\lim_{t\to T}\text{Vol}_{\omega_{t}}(M)>0$. Doing a Type-I blowup backwards in time, is the flow coming out of the singularity Cao’s expanding soliton?
In the Riemannian case, one can flow out of isolated cone singularities with positive curvature operator. Can one do this in the Kahler setting instead assuming the cones have nonnegative holomorphic bisectional curvature. -
Long-Time Existence of Complete Noncompact Kahler-Ricci Flow
Problem 3.5.
[Albert Chau] If $\sqrt{-1}\partial\overline{\partial}u$ is complete and $U(n)$-invariant Kahler metric on $\mathbb{C}^{n}$, does there exist a solution to the Kahler-Ricci flow?
This is known if $\sqrt{-1}\partial\overline{\partial}u$ has nonnegative holomorphic bisectional curvature.
Cite this as: AimPL: PDE methods in complex geometry, available at http://aimpl.org/pdecomplexgeom.