In this article, we will use the harmonic mean curvature flow to prove a new class of Alexandrov-Fenchel type inequalities for strictly convex hypersurfaces in hyperbolic space in terms of total curvature, which is the integral of Gaussian curvature on the hypersurface. We will also use the harmonic mean curvature flow to prove a new class of geometric inequalities for horospherically convex hypersurfaces in hyperbolic space. Using these new Alexandrov-Fenchel type inequalities and the inverse mean curvature flow, we obtain an Alexandrov-Fenchel inequality for strictly convex hypersurfaces in hyperbolic space, which was previously proved for horospherically convex hypersurfaces by Wang and Xia [44]. Finally, we use the mean curvature flow to prove a new Heintze-Karcher type inequality for hypersurfaces with positive Ricci curvature in hyperbolic space.

中文翻译：

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调和平均曲率流和几何不等式

在本文中，我们将使用调和平均曲率流来证明双曲空间中严格凸超曲面的一类新的 Alexandrov-Fenchel 型不等式，其总曲率是高斯曲率在超曲面上的积分。我们还将使用调和平均曲率流来证明双曲空间中水平凸超曲面的一类新几何不等式。使用这些新的 Alexandrov-Fenchel 型不等式和逆平均曲率流，我们获得了双曲空间中严格凸超曲面的 Alexandrov-Fenchel 不等式，之前 Wang 和 Xia [44] 已经证明了该超曲面的水平凸超曲面。最后，