Ricci curvature, diameter and optimal volume bound
Let \(M\) be an \(n\)-dimensional Riemannian manifold with Ricci curvature \(\text{Ric}(M) \geq (n - 1)k\) and the diameter \(d(M) \leq D\). It is known that \(M\) has volume bounded above by the volume \(\widetilde {v}_k(D)\) of a \(D\)-ball in a simply connected space form \(M^n_k\). This may be restated as follows: for the class \(M_k\) of all closed Riemann \(n\)-manifolds with \(\text{Ric} (M) \geq (n - 1)k\) and \(d(M) = \pi\) -- herewith \(k\) is naturally smaller than or equal to 1 -- the volume defines a function on \(M_k\), \(\text{vol}: M_k \to \mathbb{R}^+\), with range in the interval \((0, \widetilde{v}_k (\pi))\). To the inquiry if the upper bound \(\widetilde {v}_k(\pi)\) is optimal for \(k = 1/4\) or 1, it is shown that given any \(\Lambda \geq k(n - 1)\), there exist a function \(\Phi_\Lambda : (-\infty, 1) \to \mathbb{R}^+\) and a number \(\delta_\Lambda > 0\) such that (1) \(\Phi_\Lambda (k) \geq \widetilde{v}_k (\pi)\) and equality holds if and only if \(k = 1/4\). (2) \(\text{vol}(M) \leq \Phi_\Lambda (k)\) for any \(M \in M^\Lambda_k\), where \(M^\Lambda_k\) is a subclass of \(M_k\) defined by \(\{M \in M_k \mid \text{Ric}(M) \leq \Lambda\}\). (3) If \(k = 1/4\), then \(M \in M^\Lambda_k\) is diffeomorphic to \(\mathbb{R} P^n\) provided that \(\text{vol} (M) \geq \widetilde {v}_k (\pi) - \delta_\Lambda\).
scientific article; zbMATH DE number 800613