On minimal entropy and stability
If \((Y,g)\) is a Riemannian manifold, its volume entropy can be defined as \[ \text{Ent}(g)= \lim_{R\to\infty} R^{-1}\log\nu(R), \] where \(\nu(R)\) is the volume of a ball of radius \(R\) in the Riemannian universal covering of \(Y\). The paper under review is concerned with the minimum MinEnt for the entropy functional \(g\to\text{Ent}(g)\) and the author provides some new examples where the minimum is exactly computable (in the non-trivial case when \(\text{MinEnt}= 0\)). The previously known examples in dimension \(\geq 3\) are compact locally symmetric manifolds. The author proves (Theorem 2.6) that if \(Y\) is a closed Riemannian \(n\)-manifold whose fundamental group admits subexponential representation in the fundamental group of some closed negatively curved locally symmetric manifold \(X\) of the same dimension, then \(\text{MinEnt}(Y)^n\) is an integral multiple of \(\text{MinEnt}(X)^n\) and \(\text{MinEnt}(Y)\) is attained by some metric on \(Y\) iff \(Y\) is diffeomorphic to a covering of \(X\). Here a subexponential representation means a surjective representation with kernel of subexponential growth. The proof uses two complementary inequalities by \textit{G. Besson}, \textit{G. Courtois} and \textit{S. Gallot} [Geom. Funct. Anal. 5, No. 5, 731-799 (1995; Zbl 0851.53032)] and \textit{I. K. Babenko} [Ross. Akad. Nauk. Ser. Mat. 56, No. 4, 707-751 (1992; Zbl 0812.57022); translation in Russ. Acad. Sci., Izv., Math. 41, No. 4, 1-38 (1993)]. The conclusion follows from the results in the case of equality in the inequality in Besson, Curtois and Gallot (loc. cit.). Using Theorem 2.6, the author constructs the examples mentioned above, which have fundamental groups isomorphic to the fundamental group of a compact negatively curved locally symmetric space, by a surgery of real hyperbolic space or as a complex projective algebraic manifold.
scientific article; zbMATH DE number 1505843