Rigidity and finiteness under Ricci curvature and volume controls (Q1865184)

The author summarizes results contained in his Ph.D. thesis. Let \({\mathcal M}_{-1} (n,r_0,\varepsilon,D)\) be the class of closed \(n\)-dimensional Riemannian manifolds \(M\) with \(\text{Ric}_M\geq-(n-1)\), \(\text{diam}(M)\leq D\) and \(\text{vol-rad}_{1-\varepsilon} (M)\geq r_0\). For a complete Riemannian manifold \(M^n\) and a real number \(\varepsilon\) with \(0<\varepsilon <1\), the \((1-\varepsilon)\)-volume radius \((M)\geq r_0\) if \(\text{Vol} (B(p,r))/V_{-1} (r)\geq 1- \varepsilon\) for every \(p\in M\) and \(0<r<r_0\), where \(V_{-1}(r)\) is the volume of a geodesic ball of radius \(r\) in the hyperbolic space \(\mathbb{H}^n\) of curvature \(-1\) and \(B(p,r)\) is the geodesic ball of radius \(r\) centered at \(p\). For this class \({\mathcal M}_{-1} (n,r_0,\varepsilon,D)\) the author proves the following main result: Let \(n,r_0,D>0\) with \(n\in\mathbb{N}\), \(n\geq 3\) and \(0<r_0\leq D\). Then there exists an \(\varepsilon_0= \varepsilon_0 (n,r_0,D)\) such that for any \(M_1\) and \(M_2\in{\mathcal M}_{-1} (n,r_0,\varepsilon_0,D)\) we have that \(M_1\) and \(M_2\) are diffeomorphic provided \(\pi_1(M_1) =\pi_1(M_2)\). This means that the isomorphic types of fundamental groups characterize the diffeomorphism types of manifolds in such a class \({\mathcal M}_{-1}\). This result extends Mostow's rigidity theorem for hyperbolic \(n\)-manifolds to the class \({\mathcal M}_{-1}\). This leads to the following finiteness result: For \(n\geq 3\) and \(0<r_0\leq D\), there is an \(\varepsilon_0= \varepsilon_0 (n,r_0,D)\) such that the class \({\mathcal M}_{-1} (n,r_0, \varepsilon_0,D)\) contains only finitely many diffeomorphic types.