Gradient estimate for a locally constrained flow in $\mathbb H^{n+1}$

Problem's Description

In attempt to establish Alexandrov-Fenchel inequality for quermassintegrals in $\mathbb H^{n+1}$, Brendle-Guan-Li introduced the following hypersurface flow in $\mathbb H^{n+1}$: \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 proved for starshaped domains.

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\noindent {\bf Open problem}: prove the gradient estimate for flow (\ref{flow}).

1. Article Isoperimetric type problem and hypersurface flows

   • Pengfei Guan and Junfang Li

   Journal of Mathematical Study 56 (1), 56-80, 2021