Sur le volume minimal des variétés ouvertes. (On the minimal volume of open manifolds). (French) Zbl 0971.53027

The minimal volume of a manifold \(M\) is defined as \(\text{minvol} (M)=\inf_{|k(g)|\leq 1}\text{vol}_g(M)\), \(g\) denoting a complete Riemannian metric on \(M\) and \(k(g)\) the sectional curvature. A rigidity theorem for the minimal volume of closed manifolds is due to G. Besson, G. Curtois and S. Gallot, Geom. Funct. Anal. 5, 731-799 (1995; Zbl 0851.53032)]. In this paper the author proves that the above result cannot be extended to open manifolds; more precisely, he states the following theorem.
Theorem A. Let \(M\) be a compact, connected, orientable manifold, whose boundary is a union of \((n-1)\)-dimensional disks, \(n=\dim M\geq 3\). Assume that the interior of \(M\) admits a complete hyperbolic structure of finite volume. Then there exists a compact manifold \(N\), non-homeomorphic with \(M\), and a proper map \(f:N\to M\), with \(\deg f=1\), such that \(\text{minvol} (N) \leq\text{vol}_{\text{hyp}}(M)\). Moreover, \(N\) and \(M\) have the same simplicial volume. Following a procedure of J. Cheeger and M. Gromov, the manifold \(N\) is obtained from \(M\) by glueing on the boundary a trivial bundle \(\ddot T^2\times T^{n-2}\), where \(T^{n-2}\) is a \((n-2)\)-dimensional torus, \(\ddot T^2\) is a torus minus two disks. This construction and the rigidity theorem in [loc. cit.] allow to extend a result formulated by W. Thurston for 3-manifolds.
Theorem B. Let \(M\) be a closed, orientable \(n\)-dimensional manifold, \(n\geq 3\). Assume that there exists a hyperbolic open submanifold \(X\) of \(M,X\) complete and of finite volume, such that \(\partial\overline X\) is a finite union of tori and any connected component of \(M/X\) is homeomorphic to the product of a 2-dimensional disk with a torus. Then \(\text{vol}_{\text{hyp}}(X) >\text{vol}_{g_0}(Y)\), provided that there exists a map \(f:M\to Y\), with \((Y,g_0)\) a closed hyperbolic manifold and \(\deg f=1\).


53C20 Global Riemannian geometry, including pinching
53C24 Rigidity results


Zbl 0851.53032
Full Text: DOI Numdam EuDML


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