2. Complex Hessian Equations
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Volume Bounds for Sublevel Sets
Problem 2.1.
[Sławomir Kołodziej] It is known that given a bounded pseudoconvex domain $\Omega\subseteq\mathbb{C}^{n}$, there exist $C=C(\Omega)$ and $\alpha=\alpha(\Omega)$ such that for any $u\in PSH(\Omega)$ satisfying $u|_{\partial\Omega}\equiv0$ and $\int_{\Omega}(dd^{c}u)^{n}=1$ (and this is defined), we have \[ \int_{\Omega}e^{-\alpha u}\omega^{n}\leq C. \] A more precise estimate is \[ \text{Vol}(\{u<-s\})\leq C(\Omega)s^{n-1}e^{-2\pi s}. \]
Can one remove the $s^{n-1}$ on the right hand side? Given a compact Kahler manifold $(M,\omega)$ with boundary (and $\omega$ extends smoothly across the boundary), do these estimates generalize if instead one assume $u\in PSH(M,\omega)$, $u|_{\partial M}\equiv0$, and $\int_{\Omega}\omega^{n}=1$? That is, can one bound \[ \int_{X}e^{-\alpha\left(\varphi-\sup_{X}\varphi\right)}\omega^{n}\leq C\int_{X}\omega^{n}. \]
One could also ask this question for the complete noncompact case, with appropriately modified assumptions/conclusion. -
C^0 Bounds for Complex Hessian Equations
Problem 2.2.
[Tat Dat TÔ] Suppose $(X,\omega_{X})$ is a Kahler manifold and $\omega_{0}$ is another Kahler metric on $X$. Suppose $C_{1}<\infty$ satisfies \[ \omega_{X}\leq C_{1}\omega_{0}. \] Let $u\in C^{\infty}(X)$, and let $\lambda[u]$ be the eigenvalues of $\omega_{0}+\sqrt{-1}\partial\overline{\partial}u$ with respect to $\omega_{X}$. Consider the equation \[ F(\lambda[u])=\psi, \] where $F$ is elliptic and $\lambda[u]$ is positive. When does this satisfy an estimate of the form \[ \sup_{X}|u|\leq C(C_{1},\omega_{0},\text{Ent}_{p}(\psi)), \] where $p>n$, and \[ \text{Ent}_{p}(\psi)=\int_{X}\psi^{n}|\log\psi|^{p}\omega_{X}^{n}. \]
For example, if the $\mathcal{J}$-equation \[ \text{tr}_{\omega_{0}+\sqrt{-1}\partial\overline{\partial}u}(\omega_{X})=\text{const} \] satisfies a $C$-subsolution {[}Sz{]} property, then this is known if $n=2$ {[}SW{]}, and if $n>2$ {[}To{]}.
Two cases where this problem are interesting are: \[ \omega_{X}^{k}\wedge(\omega_{0}+\sqrt{-1}\partial\overline{\partial}u)^{n-k}=c\psi(\omega_{0}+\sqrt{-1}\partial\overline{\partial}u)^{n}, \] \[ \omega_{X}^{k}\wedge(\omega_{0}+\sqrt{-1}\partial\overline{\partial}u)^{n-k}=c\psi\omega_{X}^{n}. \] -
Gradient Estimate
Problem 2.3.
[Bin Guo] Consider the equation \[ F[\lambda[u]]=\psi, \] where $\omega_{0},\omega_{X}$ are nonnegative Hermitian $(1,1)$-forms and $\lambda[u]$ are the eigenvalues of $\omega_{0}+\sqrt{-1}\partial\overline{\partial}u$ with respect to $\omega_{X}$. Assume the $C$-subsolution condition holds, and that $F$ is concave. Is it possible to prove effective estimates (not through a contradiction argument) of the form \[ \sup_{X}|\nabla u|\leq C(\omega_{0},\omega_{X},\psi,F)? \] For example, can one prove this using maximum principle techniques?
Blocki did this for the complex Monge-Ampere equation using maximum principle techniques, but so far there is no known direct proof in the case of the $\sigma_{k}$ equations. It is also known in the case of deformed Hermitian-Yang-Mills. -
Liouville Property
Problem 2.4.
Suppose $\det(\partial_{i}\partial_{\overline{j}}u)=1$ on $\mathbb{C}^{n}$ and \[ C^{-1}(1+|z|^{2})\leq\partial_{i}\partial_{\overline{j}}u(z)\leq C(1+|z|^{2}) \] for all $z\in\mathbb{C}^{n}$. Is it true that $u$
It known if, in addition, $\sqrt{-1}\partial\overline{\partial}u$ is a complete Kahler metric on $\mathbb{C}^{n}$.
It is known if $u(z)=\langle Az,\overline{z}\rangle+o(|z|)$ as $|z|\to\infty$ (Yu Wang 2012).
Is there a way to use the Kahler-Ricci flow to solve this problem? Is the Kahler-Ricci flow starting at $\sqrt{-1}\partial\overline{\partial}u$ instantaneously complete?
Suppose $u$ is such that $\sqrt{-1}\partial\overline{\partial}u$ is complete, close to the Taub-Nut (maybe in the sense that $u(z)-P(z)=o(|z|)$, where $P$ is the Kahler potential for the Taub NUT metric) and is Ricci flat.
Cite this as: AimPL: PDE methods in complex geometry, available at http://aimpl.org/pdecomplexgeom.