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Gauss-Bonnet-Chern mass and Alexandrov-Fenchel inequality

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Abstract

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.

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118, route de Narbonne, 31062, Toulouse Cedex, France

    Yuxin Ge

  2. Albert-Ludwigs-Universität Freiburg, Mathematisches Institut Eckerstr, 1, D-79104, Freiburg, Germany

    Guofang Wang

  3. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

    Jie Wu

  4. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

    Chao Xia

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  1. Yuxin Ge
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  2. Guofang Wang
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  3. Jie Wu
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  4. Chao Xia
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Correspondence to Guofang Wang.

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Ge, Y., Wang, G., Wu, J. et al. Gauss-Bonnet-Chern mass and Alexandrov-Fenchel inequality. Front. Math. China 11, 1207–1237 (2016). https://doi.org/10.1007/s11464-016-0558-3

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