Open
Posted online: 2022-10-12 18:55:53Z by Pengfei Guan13
Cite as: P-221012.2
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).
\medskip
\noindent {\bf Open problem}: prove the gradient estimate for flow (\ref{flow}) $\forall k< n-1$.
\medskip
The case $k=n-1$ is known, as the flow preserved convexity.
No solutions added yet
Edited: (general update ) at 2022-10-12 19:08:36Z
Created at: 2022-10-12 18:55:53Z View this version
No remarks yet