Some Local and Non-local Variational Problems in Riemannian Geometry

— In this article we will give a brief summary of some recent work on two variational problems in Riemannian geometry. Although both involve the study of elementary symmetric functions of the eigenvalues of the Ricci tensor, as far as technique and motivation are concerned the problems are actually quite different. Résumé (Problèmes variationnels locaux et non-locaux en géométrie riemannienne) Dans cet article nous donnons un aperçu d’un travail récent sur deux problèmes variationnels en géométrie riemannienne. Bien que les deux problèmes soient basés sur l’étude des fonctions symétriques élémentaires des valeurs propres du tenseur de Ricci, les techniques et les motivations sont en réalité différentes. For since the fabric of the universe is most perfect and the work of a most wise Creator, nothing at all takes place in the universe in which some rule of maximum or minimum does not appear. –Leonhard Euler 1. Quadratic Riemannian functionals The first problem we will discuss represents joint work of the author with Jeff Viaclovsky ([GV00]). To describe it, let us begin with some general notions. Let M be a smooth manifold, M the space of smooth Riemannian metrics on M , and G the diffeomorphism group of M . A functional F : M → R is called Riemannian if F is invariant under the action of G; i.e., if F (φ∗g) = F (g) for each φ ∈ G and g ∈ M. If we endow M with a natural L–Sobolev norm, then we may speak of differentiable Riemannian functionals. Letting S2(M) denote the bundle of symmetric two–tensors, we then say that F : M → R has a gradient at g ∈ M if d dt F [g + th]|t=0 = ∫ g(h,∇F ) d volg for some ∇F ∈ Γ(S2(M)) and all h ∈ Γ(S2(M)). 2000 Mathematics Subject Classification. — 53Cxx, 58Jxx.