Gauss-Bonnet-Chern mass and Alexandrov-Fenchel inequality
Gauss-Bonnet-Chern mass and Alexandrov-Fenchel inequality
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DOI：
10.1007/s11464-016-0558-3
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发表时间：
2016-08
影响因子：
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通讯作者：
Yuxin Ge;Guofang Wang;Jie Wu;C. Xia
中科院分区：
文献类型：
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作者：
Yuxin Ge;Guofang Wang;Jie Wu;C. Xia
本文综述了我们最近关于渐近平坦和渐近双曲流形的gaus - bonnet - chern （GBC）质量的研究工作。我们首先利用高阶标量曲率分别引入渐近平面流形和渐近双曲流形的高阶质量GBC质量。然后证明了它的正性和图形流形的Penrose不等式。证明彭罗斯不等式的关键步骤之一是使用亚历山德罗夫-芬切尔不等式，它是欧几里德空间中的一个经典不等式。在双曲空间中，我们建立了新的Alexandrov-Fenchel不等式。对于渐近局部双曲流形，我们也有类似的工作。最后讨论了GBC质量与陈氏魔形之间的关系。
This is a survey about our recent works on the Gauss-Bonnet-Chern (GBC) mass for asymptotically flat and asymptotically hyperbolic manifolds. We first introduce the GBC mass, a higher order mass, for asymptotically flat and for asymptotically hyperbolic manifolds, respectively, by using a higher order scalar curvature. Then we prove its positivity and the Penrose inequality for graphical manifolds. One of the crucial steps in the proof of the Penrose inequality is the use of an Alexandrov-Fenchel inequality, which is a classical inequality in the Euclidean space. In the hyperbolic space, we have established this new Alexandrov-Fenchel inequality. We also have a similar work for asymptotically locally hyperbolic manifolds. At the end, we discuss the relation between the GBC mass and Chern’s magic form.