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Posted online: 2022-10-12 18:55:53Z by Pengfei Guan

Cite as: P-221012.2

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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}).

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