An excess sphere theorem

Abstract

An excess sphere theorem Annales scientifiques de l'É.N.S. 4 e série, tome 26, n o 2 (1993), p. 175-188 <http://www.numdam.org/item?id=ASENS_1993_4_26_2_175_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1993, tous droits réservés. L'accès aux archives de la revue « Annales scientifiques de l'É.N.S. » (http://www. elsevier.com/locate/ansens) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Smoothing Riemannian metrics with Ricci curvature bounds

“…With (5.1) and ( 5.3) the proof of Theorem 1.4 is quite similar. We observe that the original statement in [PZ,Theorem 2.1] can be restated as that there exists an ǫ(n) such that if M n is a compact manifold satisfying |K M | ≤ 1, ex(M)/diam(M) ≤ ǫ(n), then either M is infranil or diffeomorphic to the union of two normal bundles over two embedded infranilmanfolds in M. We then follow the above argument.…”

Smooth stability and sphere theorems for manifolds and Einstein manifolds with positive scalar curvature

“…In view of Grove-Shiohama's celebrated diameter sphere theorem for positive sectional curvature (see [27]) and the wealth of other sphere theorems for manifolds of positive sectional and of positive Ricci curvature (see, e.g., [4], [31], [14], [37], [2], [8], [18], [22], [26], [29], [33], [34], [36], [39], [42], [44]), it is therefore natural to ask which conditions on the injectivity radius, or, more generally, conjugate radius, of a closed Riemannian n-manifold M with positive scalar curvature will guarantee stability of Green's above-mentioned results in the sense that M can still be recognized as being homeomorphic, or even diffeomorphic, to the standard n-sphere or, respectively, to an n-dimensional spherical space form.…”

Curvature, triameter, and beyond

“…For this reason it is not only natural but necessary to pursue and investigate other metric invariants. Recently such investigations have included Gromov's filling radius (cf., e.g., [G,Kl,K2,W]), excess invariants (cf., e.g., [GP2,O,PZ]), and Urysohn's intermediate diameters (cf., e.g., [U, G, K3]). …”