the Eells-Sampson theorem
/ EELZ SAMP-sun /
You have a continuous map between two curved spaces and you want to replace it by the 'best' map in its homotopy class — the most relaxed, least-stretched one. Can you always do that? The Eells-Sampson theorem says yes, provided the target space curves the right way: if the target is nonpositively curved, every homotopy class of maps from a compact manifold contains a harmonic representative. It is the existence theorem that made harmonic-map theory a working tool rather than a wish.
Precisely: let M be a compact Riemannian manifold and N a compact Riemannian manifold with nonpositive sectional curvature. Then any smooth map f_0: M -> N is homotopic to a smooth harmonic map, and that map minimizes Dirichlet energy in its homotopy class. The proof, due to James Eells and Joseph Sampson in 1964, introduced the harmonic-map heat flow: evolve the map by partial_t f = tau(f), the gradient flow of the Dirichlet energy, so energy decreases along the flow. The heart of the argument is a Bochner-type identity showing that under nonpositive target curvature the energy density satisfies a differential inequality (a parabolic maximum principle) that prevents blow-up; the flow therefore exists for all time and subconverges to a harmonic map. The nonpositive curvature of the target is what makes the dangerous curvature term in the Bochner formula have a good sign — that single hypothesis is doing all the work.
The theorem is a landmark for two reasons: it solved a hard existence problem by a parabolic-flow method that became the template for Hamilton's Ricci flow twenty years later, and its harmonic maps became rigidity tools (Siu's and Mostow-type rigidity, super-rigidity) where the nonpositive-curvature hypothesis is natural. The honest caveats are essential. The nonpositive curvature of the target is not a technical convenience but a genuine necessity: into positively curved targets the conclusion fails — there are homotopy classes of maps into spheres with no harmonic representative, and the heat flow can develop finite-time singularities and bubble off energy. Also 'subconverges' is honest wording: the flow converges along a subsequence (and to a unique limit when the target is negatively curved and the map is not into a closed geodesic), not necessarily for the full time-limit without extra hypotheses.
Any continuous map between two compact hyperbolic surfaces (both have constant negative curvature, hence nonpositive) is homotopic to a unique harmonic map; this is the starting point of Schoen-Yau and Sampson rigidity arguments comparing hyperbolic structures.
Between negatively curved targets the harmonic representative is unique — the cleanest form of the theorem.
Eells-Sampson is false for positively curved targets: there is no harmonic map of degree one from the torus to the sphere, so 'every map can be made harmonic' is true only under the nonpositive-curvature hypothesis, never in general.