On minimal entropy and stability
type: Item
instance of: scholarly article
title: On minimal entropy and stability
zbMATH Open document ID: 1010.53033
DOI: 10.1023/A:1005247027248
author: Andrea Sambusetti
published in: Geometriae Dedicata
publication date: 2003-05-14T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 37D35
zbMATH DE Number: 1505843
zbMATH Keywords: minimal entropy
zbMATH Keywords: locally geometric compact manifolds
zbMATH Keywords: subexponential representation
zbMATH Keywords: surgery
reviewed by: Gueo Grantcharov
MaRDI profile type: Publication
full work available at URL: a:1005247027248
OpenAlex ID: W103029006
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description: scientific article; zbMATH DE number 1505843