×

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 \(\deg f=1\).

MSC:

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

Citations:

Zbl 0851.53032
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] L. BESSIÈRES, Un théorème de rigidité différentielle, Commentarii Mathematici Helvetici, vol. 73, juin (1998). · Zbl 0909.53024
[2] G. BESSON, G. COURTOIS et S. GALLOT, Entropies et rigidités des espaces localement symétriques de courbure strictement négative, GAFA, vol. 5 (5), octobre 1995. · Zbl 0851.53032
[3] R. L. BISHOP, B. O’NEILL, Manifolds of negative curvature, Trans. of the A.M.S., vol. 145, november (1969). · Zbl 0191.52002
[4] J. CHEEGER, M. GROMOV, Collapsing Riemannian manifolds while keeping their curvature bounded I, J. Differential geometry, 23 (1986), 309-346. · Zbl 0606.53028
[5] M. GROMOV, Volume and bounded cohomology, IHES, 56 (1981). · Zbl 0516.53046
[6] W. JACO, Lectures on 3-manifolds topology, Amer. Math. Soc., 43 (1980). · Zbl 0433.57001
[7] W. THURSTON, The geometry and topology of 3-manifolds, Princeton University Press, Princeton, 1978.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.