SURVEY ARTICLE

Gauss-Bonnet-Chern mass and Alexandrov-Fenchel inequality

  • Yuxin GE 1 ,
  • Guofang WANG , 2 ,
  • Jie WU 3 ,
  • Chao XIA 4
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  • 1. Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex, France
  • 2. Albert-Ludwigs-Universität Freiburg, Mathematisches Institut Eckerstr, 1 D-79104 Freiburg, Germany
  • 3. Department of Mathematics, Zhejiang University, Hangzhou 310027, China
  • 4. School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

Received date: 16 Feb 2016

Accepted date: 20 Jun 2016

Published date: 23 Sep 2016

Copyright

2016 Higher Education Press and Springer-Verlag Berlin Heidelberg

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.

Cite this article

Yuxin GE , Guofang WANG , Jie WU , Chao XIA . Gauss-Bonnet-Chern mass and Alexandrov-Fenchel inequality[J]. Frontiers of Mathematics in China, 2016 , 11(5) : 1207 -1237 . DOI: 10.1007/s11464-016-0558-3

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