Total mean curvatures of Riemannian hypersurfaces
Total mean curvatures of Riemannian hypersurfaces
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DOI：
10.1515/ans-2022-0029
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发表时间：
2022-04
影响因子：
1.8
通讯作者：
M. Ghomi;J. Spruck
中科院分区：
文献类型：
--
作者：
M. Ghomi;J. Spruck
摘要利用Reilly恒等式得到了Riemannian超曲面平均曲率积分的比较公式。作为应用，我们得到了Cartan-Hadamard流形M M中凸超曲面Γ\Gamma的几个几何不等式.特别地，我们证明了嵌套在Γ\Gamma内的凸超曲面γ\Gamma的第一平均曲率积分不能超过Γ\Gamma的第一平均曲率积分，这导致了Γ\Gamma的总第一平均曲率的一个尖锐的下界，根据它在3维M M中的体积。当γ\Gamma平行于Γ\Gamma或M M具有常曲率时，这种单调性推广到所有平均曲率积分.我们还刻画了Cartan-Hadamard流形中等半径球之间的平均曲率积分的极小化的双曲球。
Abstract We obtain a comparison formula for integrals of mean curvatures of Riemannian hypersurfaces via Reilly’s identities. As applications, we derive several geometric inequalities for a convex hypersurface Γ \Gamma in a Cartan-Hadamard manifold M M . In particular, we show that the first mean curvature integral of a convex hypersurface γ \gamma nested inside Γ \Gamma cannot exceed that of Γ \Gamma , which leads to a sharp lower bound for the total first mean curvature of Γ \Gamma in terms of the volume it bounds in M M in dimension 3. This monotonicity property is extended to all mean curvature integrals when γ \gamma is parallel to Γ \Gamma , or M M has constant curvature. We also characterize hyperbolic balls as minimizers of the mean curvature integrals among balls with equal radii in Cartan-Hadamard manifolds.