This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is to establish geometric convergence of a (sub)sequence of manifolds with boundary with such geometrical bounds and also an upper bound on the diameter and a lower bound on injectivity and boundary injectivity radius, making use of the first part. The third theme involves the uniqueness and conditional stability of an inverse problem proposed by Gel'fand, making essential use of the results of the first two parts.

L'accès aux archives de la revue « Publications mathématiques de l'I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) 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/ CLOSED ORBITS IN HOMOLOGY CLASSES by ATSUSHI KATSUDA and TOSHIKAZU SUNADA Dedicated to Pr. Akihiko Morimoto for his 60th birthday 6 ATSUSHI KATSUDA AND TOSHIKAZU SUNADA hence the winding cycle is regarded as the average of the c< homological " direction in which the orbits are traveling. The central limit theorem (cf. Denker and Philipp [5]) guarantees the existence of the limit 8( (9^ x) rfr-^O(co)) 2 , v •/x \Jo / which yields a positive semi-definite quadratic form on H^X, R). We call 8 the cornnance form. As we will see later, 8 is positive definite on Ker 0, and hence gives rise to a Euclidean metric on Ker 0. Consider the character group H ofH. The tangent space T^ H at the trivial character 1 is identified with the dual H 1 ' == Hom(H,R), which is also identified, in a natural manner, with a subspace in H^X,^. Therefore if 0 vanishes on H 1 ', the covariance form induces a flat metric on the group H. We denote by vol(H) the volume with respect to the metric. Theorem 1 (Density theorem).-If 0 vanishes on the dual H 1 ', then (0.1) 7r(^a)-G^^ asx^co, where b == rank H and C == {2^)-^ vol(H)-1 h~\ The above condition on the winding cycle is necessary for the asymptotic like (0.1). In fact we have the following Theorem 2.-7/'

No abstract

No abstract