Hyperbolic Alexandrov-Fenchel quermassintegral inequalities I

By Yuxin Ge, Guofang Wang, Jie Wu. 2013

49 citations recorded by the source corpus.

In this paper we prove the following geometric inequality in the hyperbolic space $\H^n$ ($n\ge 5)$, which is a hyperbolic Alexandrov-Fenchel inequality, \[\begin{array}{rcl} \ds \int_\Sigma \s_4 d \mu\ge \ds\vs C_{n-1}^4\omega_{n-1}\left\{\left(\frac{|\Sigma|}{\omega_{n-1}} \right)^\frac 12 + \left(\frac{|\Sigma|}{\omega_{n-1}} \right)^{\frac 12\frac {n-5}{n-1}} \right\}^2, \end{array}\] provided that $\Sigma$ is a horospherical convex hypersurface. Equality holds if and only if $\Sigma$ is a geodesic sphere in $\H^n$.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b11037a7bfb549f0bdb3907e600bd4606ded9ba5

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In Applied Mathematics

1. Guofang Wang
2. Jie Wu
3. Yuxin Ge
4. Y. Yang
5. Chao Xia

In Geometry and Topology

1. Guofang Wang
2. Jie Wu
3. Yuxin Ge

Citations in the corpus

• Rigidity results, inverse curvature flows and Alexandrov-Fenchel type inequalities in the sphere (2013, unknown): “Some of these inequalities have also been shown in hyperbolic space by using inverse F -curvature flows, see for example [7], [8], [21].”
• Hyperbolic Alexandrov-Fenchel quermassintegral inequalities II (2013, background): “ second fundamental form. Proof. For the long time existence of the inverse curvature flow, see the work of Gerhardt [16]. The preservation of the horospherical convexity along flow (2.7) was proved in [15] with the help of a maximal principle for tensors of Andrews [1] .8 YUXIN GE, GUOFANG WANG, AND ”