Simplices of maximal volume in hyperbolic n-space

An obstruction to the existence of Einstein metrics on 4-manifolds

“…Recall that for non-orientable manifolds Gromov's theorem 1.2 still holds, while Thorpe's theorem takes a simpler form, previously proved by M. Berger, which asserts that a compact 4-manifold Y has no Einstein metric if χ(Y ) < 0, and that, if χ(Y ) = 0, Y has no Einstein metric unless it is flat. Theorem 2.4 also gives in some cases an improvement of Gromov's inequality (Theorem 1.2) which can be written in the following way: Let T 4 be the volume of the regular 4-dimensional ideal geodesic simplex in the real hyperbolic 4-dimensional space: it is explicitly computable and its value is (see [10] )…”

Scissors Congruence, the Golden Ratio and Volumes in Hyperbolic 5-Space

“…[4] and [1]) while μ 5 is the maximal volume among all hyperbolic 5-simplex volumes (cf. [6] and [13]). Of course, all three constants are certain trilogarithmic expressions.…”

“…A first analytical proof, for arbitrary dimension n, is due to U. Haagerup and H.J. Munkholm [6], a second and geometrical proof based on Steiner symmetrisation is due to N. Peyerimhoff [13]. This extremality property is a key, for example, in Gromov's proof [5] of Mostow rigidity.…”

The proportionality constant for the simplicial volume of locally symmetric spaces

“…As an application of our method, let us give an elementary proof of Gromov's result that the proportionality constant for hyperbolic n-manifolds is v n = sup{|Vol(σ)| | σ : ∆ n → H n a geodesic simplex} = |volume of the regular ideal geodesic simplex in H n |, where the last equality follows from [HaMu81]. Let thus G denote the group of orientation preserving isometries of the n-dimensional hyperbolic space H n .…”