Rigidity of nonpositively curved graph manifolds
Let M be a connected 3-dimensional manifold without boundary. M is called an NP-graphmanifold if M is a finite or countable union of ''building blocks'' \(M_ i\) satisfying the following: I) Every building block \(M_ i\) is diffeomorphic to \(W_ i\times S^ 1\), where \(W_ i\) is topologically a compact surface with a finite number of disjoint balls removed and is neither a disk nor an annulus. 2) If \(T_ 1\) is a boundary torus in a building block \(M_ 1\), then there exists exactly one other building block \(M_ 2\) and a boundary torus \(T_ 2\) such that \(M_ 1\) and \(M_ 2\) are glued together by a diffeomorphism of \(T_ 1\) onto \(T_ 2\) that interchanges the \(S^ 1\) factors. \textit{M. Gromov} showed [J. Differ. Geom. 13, 223-230 (1978; Zbl 0433.53028)] that given a constant \(\epsilon >0\) and an NP-graphmanifold M there exists a metric \(g_{\epsilon}\) on M such that \(volume(M,g_{\epsilon})\leq \epsilon\) and the sectional curvature K satisfies -1\(\leq K\leq 0\). The metric of nonpositive curvature on each building block \(M_ i=W_ i\times S^ 1\) is a product in a neighborhood of the boundary of \(M_ i\) and is uniquely determined up to scaling by a homomorphism \(\rho\) : \(\pi\) \({}_ 1(w_ 1)\to ({\mathbb{R}},+)\). The NP-graphmanifolds give a counterexample in the \(C^{\infty}\) category to the conjecture that for every integer \(n\geq 3\) there exists a positive constant \(c_ n\) such that \(volume(M_ n,g)\geq c_ n\) for every n-manifold \(M_ n\) whose universal cover \(\tilde M_ n\) has no Euclidean factor and whose sectional curvature satisfies -1\(\leq K\leq 0\). (This conjecture is true for real analytic manifolds - see section 12 of \textit{W. Ballman}, \textit{M. Gromov} and the author, ''Manifolds of nonpositive curvature'' (1985; Zbl 0591.53001)]. The main result of the paper is that one can identify an NP-graphmanifold from its fundamental group. Theorem. Let \(M^*\) be an NP-graphmanifold and let M be a complete, 3-dimensional Riemannian manifold of nonpositive sectional curvature such that \(\pi_ 1(M)=\pi_ 1(M^*)\). Then M is metrically a graphmanifold and is diffeomorphic to \(M^*\) by a map that carries building blocks of M onto building blocks of \(M^*\).
scientific article; zbMATH DE number 3949228