On the geometric flows solving Kählerian inverse \(\sigma_k\) equations (Q453224)

Suppose \(\left( M^{n},\omega \right) \) is a compact Kähler manifold. The authors consider the geometric curvature flow and in particular solutions to the inverse \(\sigma _{k}\) problem which is a fully nonlinear geometric PDE. They show that there is a large set of nonlinear geometric flows which converge to the solution of the inverse \(\sigma _{k}\) problem.NEWLINENEWLINELet \(\chi \) be a Kähler metric in the cohomology class \(\left[ \chi \right] .\) For a fixed integer \(k\in \left[ 1,n\right] \) define \(\sigma _{k}\left( \chi \right) =\left( _{k}^{n}\right) \frac{\chi ^{k}\wedge \omega ^{n-k}}{\omega ^{n}}.\) If \(c_{k}:=\frac{\int_{M}\sigma _{n-k}\left( \chi \right) }{\int_{M}\sigma _{n}\left( \chi \right) }=\left( _{k}^{n}\right) \frac{\left[ \chi \right] ^{n-k}\left[ \omega \right] ^{k}}{\left[ \chi \right] ^{n}}\), the problem is to determine if there exists a metric \( \widetilde{\chi }\in \left[ \chi \right] \) satisfying \(c_{k}\widetilde{\chi }^{n}=\left( _{k}^{n}\right) \widetilde{\chi } ^{n-k}\wedge \omega ^{k}\) (*). The authors consider a general flow of the form \(\frac{\partial }{\partial t} { \varphi =F}\left( \chi _{\varphi }\right) { -f}\left( c_{k}\right) \) for \(f\in C^{\infty }\left( \mathbb{R}_{>0},\mathbb{R}\right) \) in the space \(\mathcal{P}_{\chi }:=\left\{ \varphi \in C^{\infty }\left( M\right) \left| \chi _{\varphi }:=\chi +\frac{i}{2}\partial \overline{ \partial }\varphi >0\right. \right\} ,\) where \({ F}\left( \chi _{\varphi }\right) =f\left[ \frac{\sigma _{n-k}\left( \chi _{\varphi }\right) }{\sigma _{n}\left( \chi _{\varphi }\right) }\right]\). The main theorem is that under these hypotheses with \(\left[ \chi \right] \) satisfying a Kähler cone condition, if \(f^{\prime }<0\), \(f^{\prime \prime }\geq 0,\) and \(f^{\prime \prime }+\frac{f^{\prime }}{x}\leq 0,\) then the flow with any initial value \(\chi _{0}\in \left[ \chi \right] \) has long time existence and the metric \(\chi _{\varphi }\) converges in the \(C^{\infty }\) norm to the critical metric \(\widetilde{\chi }\in \left[ \chi \right] \) and unique solution of (*).