A theorem on differential rigidity (Q1266445)

Let \(M,N\) be two smooth connected closed orientable manifolds of the same dimension \(n\geq 3\), such that \(M\) is endowed with a hyperbolic metric \(g^0\) of sectional curvature \(K_{g^0} =-1\). Theorem 1 here states that if \(f:N\to M\) is a continuous map of degree \(p\geq 1\) such that \(\in\{\text{vol}_g(M):g\) is a Riemannian metric with sectional curvature \(| K(g) |\leq 1\} =p \text{vol}_{g^0}(M)\), then \(N\) is hyperbolic and \(f\) is homotopic to a lift of degree \(p\). This result generalizes a theorem due to Thursten on the application of degree \(p\) between two connected orientable closed hyperbolic manifolds. The proof of Theorem 1 uses [\textit{M. Gromov}, `Structures métriques pour les variétés riemanniennes' (Textes Math. 1, Cedic/Fernand Nathan, Paris) (1981; Zbl 0509.53034)] on the Riemannian convergence in the sense of Gromov as well as [\textit{G. Besson}, \textit{G. Courtois} and \textit{S. Gallot}, Geom. Funct. Aral. 5, 731-799 (1975; Zbl 0851.53032)], where it is shown that on a closed hyperbolic manifold, the minimal volume is reached by a hyperbolic metric. An interesting application of Theorem 1 states that the minimal volume is not an invariant of topological type of the manifold, but only of its differential structure. Moreover, the minimal volume is not additive on the connected sum.