On the minimal volume of open manifolds (Q2709135)

Let \(N\) be an open \(n\)-dimensional (\(n\geq 3\)) differential manifold, The minimal volume of \(N\) is defined as \(\text{Minvol}(N)=\inf_{|K(g)|\leq 1} \text{vol}_{g}(N)\) , where \(g\) is a complete Riemannian metric on \(N\) and \(K(g)\) is the sectional curvature. If \(\;N\) is compact with border, one considers complete metrics on the interior. It is known that if \(N\) and \((M,g_{0})\) are closed, connected oriented manifolds of the same dimension (greater or equal to 3), \(g_{o}\) is an hyperbolic metric and there is a continuous map \(f:N\rightarrow M\) with degree \(\deg (f)\neq 0;\) then (1) \(\text{Minvol}(N)\geq |\deg (f)|\text{vol}_{g_{0}}(M)\). Furthermore, equality is obtained if and only if \(N\) has a hyperbolic metric and there is a differentiable covering\ from \(N\) to \(M\) homotopic to \(f\) [see \textit{G. Besson, G. Courtois} and \textit{S. Gallot}, Geom. Funct. Anal. 5, 731-799 (1995; Zbl 0851.53032) and \textit{L. Bessières}, Comment. Math. Helv. 73, 443-479 (1998; Zbl 0909.53024)]. NEWLINENEWLINENEWLINEHyperbolic metrics play an important role for the minimal volume. If the manifold has a hyperbolic metric the minimal volume is attained by that metric (see the above mentioned paper by \textit{G. Besson, G. Courtois} and \textit{S. Gallot}, for the closed and for the open manifold see \textit{J. Boland, C. Connel} and \textit{J. S. Clement} in [``The minimal volume of open rank one symmetric manifolds'', preprint]). NEWLINENEWLINENEWLINEThese results suggest the study of a similar problem for open manifolds. The solution of the problem is given in this article. The author proves that if \(\;M\) is an \(n\)-dimensional (\(n\geq 3\)) compact connected oriented manifold with a complete hyperbolic structure of finite volume, in its interior of finite volume, and such that its border is the union of tori \(T^{n-1}\), then there exists a compact connected manifold with border \(N\), non homeomorphic to \(M\) and a proper map \(f:N\rightarrow M\) with degree \(\deg (f)=1\) whose restriction to the border is a homeomorphism and such that (2) \(\text{Minvol}(N)\leq \text{vol}_{\text{hyp}}(M)\). Since the equality in (1) holds for open manifolds (see the preprint of \textit{J. Boland, C. Connel} and \textit{J. S. Clement} mentioned above) we will have equality in (2). NEWLINENEWLINENEWLINEThe rigidity theorem used to prove this result allows for the generalization to any dimension of a result obtained by \textit{W. Thurston} in his notes [``The geometry and topology of 3 manifolds'', Princeton (1978; Zbl 0873.57001)]. Let \(M\) be an \(n\)-dimensional (\(n\geq 3\)) closed oriented manifold. Let us suppose there is an open complete hyperbolic submanifold \(X\) of \(M\), with finite volume, such that \(\partial X=\bigcup _{i=1}^{p}T_{i}^{n-1}\) and each connected component of \(M\backslash X\) is homeomorphic to \(D^{2}\times T^{n-2}\). If \(M\) dominates a closed hyperbolic manifold \((Y,g_{0})\) by a map of degree 1, then \(\text{vol}_{\text{hyp}}(X)> \text{vol}_{g_{0}}(Y)\).NEWLINENEWLINEFor the entire collection see [Zbl 0955.00015].