A REPORT ON HARMONIC MAPS J. EELLS AND L. LEMAIRE 1. Introduction (1.1) A map between Riemannian manifolds is harmonic if the divergence of its differential vanishes. (Those terms will be defined in §3.) Such maps are the extrema ( = critical points) of the energy functional. More precisely, if <$>: M -*• N is a map between Riemannian manifolds, we define its energy by the formula M where d(f>(x) denotes the differential of 0 at the point xeM; and dx is the volume element of M. The Euler-Lagrange operator associated with E shall be written x(4>) = div (dcj)) and called the tension field of<j>. A map is said to be harmonic if its tension field vanishes identically. In physical terms, we imagine M made of " rubber " and N of marble; the map ^ constrains M to lie on N. Then with each point xeM we have a vector T(</>)(*) = div (d(j>(x)) at the point <f)(x)eN, representing the tension in the " rubber " at that point. Thus <j> is harmonic if and only if <j> constrains M to lie on TV in a position of elastic equilibrium. (1.2) Harmonic maps appear in many different contexts; e.g., (a) If dim M = 1, then the harmonic maps are the geodesies of N. (b) If JV = R, they are the harmonic functions on M. (c) If N = S l (= the unit circle), then the harmonic maps are canonically identified with the harmonic 1-forms on M with integral periods. (d) If dim M = 2, they include (parametric representations of) the minimal surfaces of JV; the energy is the Dirichlet-Douglas integral. (e) If M and N are Kahler manifolds, then holomorphic maps of M into N are harmonic, with respect to any compatible metrics. (f) If M is a Riemannian submanifold of N of minimal volume, then the inclusion map i: M -> N is harmonic. One object of this report is to display a wide variety of specific examples of harmonic maps, as they arise in various branches of mathematics. Existence problems (1.3) The basic existence problem for harmonic maps can be formulated in the following manner: Let (f> 0 : M -> N be a map of Riemannian manifolds. Can <f> 0 be deformed into a harmonic map (f>: M -*• N ? Received December 1977. PULL. LONDON MATH. SOC, 10 (1978), 1-68]