The following hypersurface flow in $\mathbb H^{n+1}$ was introduced by Brendle-Guan-Li:
\begin{align}
X_t=\Big(\cosh(\rho) \frac{\sigma_{k}}{\sigma_{k+1}}(\kappa)-\frac{u}{c_{n,k}}\Big)\nu,
\label{flow}
\end{align}
where $\nu$ the outer normal, $\kappa$ the principal curvature vector. The flow preserves $k$-th quermassintegral $\mathcal{A}_{k}$ and decreases $k+1)$-th quermassintegral $\mathcal{A}_{k+1}$. All regularity estimates and convergence would follow if one can establish the gradient estimate (or preservation of starshapedness) for solution of the flow. In turn, Alexandrov-Fenchel inequality for quermassintegrals in hyperbolic space can be established for general starshaped $k+1$-domains.

When $k=0$, Alexandrov-Fenchel inequality is the Minkowski inequality. This inequality is proved by Brendle-Guan-Li for general mean convex starshaped domains in $\mathbb H^{n+1}$. Alexandrov-Fenchel inequality is known for horo-convex domains (Wang-Xia).

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\noindent {\bf Open problem}: prove the gradient estimate for flow (\ref{flow}) $\forall k< n-1$.

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The case $k=n-1$ is known, as the flow preserved convexity.

Article
Isoperimetric type problem and hypersurface flows

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