Gauss-Bonnet-Chern mass and Alexandrov-Fenchel inequality

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.

Key words： Gauss-Bonnet-Chern (GBC) mass; Gauss-Bonnet curvature; positive mass theorem (PMT); asymptotically hyperbolic manifold; Penrose inequality; Alexandrov-Fenchel inequality

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