On the extendability of the Kneser–Poulsen conjecture to Riemannian manifolds

Abstract

Abstract The Kneser–Poulsen conjecture claims that if some balls of the Euclidean space are rearranged in such a way that the distances between the centers do not increase, then the volume of the union of the balls does not increase. Though the conjecture is still open in dimensions ≥ 3, important special cases of it have been verified not only in the Euclidean space but also in the hyperbolic and spherical spaces, raising the question about the widest class of Riemannian manifolds in which the conjecture can hold. Our main result is that if the conjecture is true for 3 balls in a complete Riemannian manifold, then the manifold must be of constant curvature, and if, furthermore, the manifold is connected, then it must be simply connected.