On the filling radius of positively curved manifolds (Q806073)

The paper establishes the following analogs of Bonnet's theorem, Topogonov's maximal diameter theorem, and the differentiable diameter sphere conjecture with Gromov's filling radius in place of the diameter function. Main Theorem 1. Let \(S^ n\) denote the unit sphere in \({\mathbb{R}}^{n+1}\), and let \({\mathcal M}\) denote the class of closed Riemannian n-manifolds with sectional curvature \(\geq 1\). For all \(M\in {\mathcal M}\), 1. Fill Rad(M)\(\leq Fill Rad(S^ n)\). 2. If Fill Rad(M)\(=Fill Rad(S^ n)\), then M is isometric to \(S^ n\). 3. There is a \(\delta\) (n)\(\geq 0\) so that if Fill Rad(S\({}^ n)-\delta (n)<Fill Rad(M)\), then M is diffeomorphic to \(S^ n.\) Motivated by results of M. Katz, the following metric invariant is defined: the spread of a compact metric space, X, is the smallest number \(R>0\) so that there is a weak R-net \(Y\subset X\) with diameter(Y)\(\leq R\). The main theorem is a corollary of a similar result in which the statements about filling radius are replaced by analogous statements about spread. The theorem about spread follows from a result of Yamaguchi's Gromov-Hausdorff convergence and the following result. Theorem 2. If X is in the Gromov-Hausdorff closure of \({\mathcal M}\), then either X is isometric to \(S^ n\), or \(Spread(X)<Spread(S^ n).\) The strategy for proving Theorem 2 is to mimic an algorithm for constructing the set of vertices of a regular inscribed \((n+1)\)-simplex in \(S^ n\), and to pass the construction from a convergent sequence \(\{M_ i\}\subset {\mathcal M}\) to its limit X. This is accomplished using convexity ideas (à la Cheeger-Gromoll and Greene-Wu) and by controlling critical points for smooth approximations of distance functions (à la Grove-Shiohama).