## 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

Zbl 0851.53032
Full Text:

### References:

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