Eells–Sampson theorem

Remarks

The key estimate is a Bochner formula for the energy density $e = \frac{1}{2} \abs{du}^2$ along the flow: $(\partial_t - \Delta_g) e \le C e$, where $C$ depends on a lower bound for the Ricci curvature of $M$. The curvature term of $N$ enters with a sign that helps because the sectional curvature of $N$ is nonpositive. Together with the decrease of the energy, this gives uniform bounds on $e$, hence on all derivatives by parabolic regularity, so the flow cannot blow up. Since $\int_0^\infty \int_M \abs{\tau(u)}^2 \, dV_g \, dt \le E(u_0)$, the tension tends to zero along a sequence of times, and compactness gives the harmonic limit.

Example: for $M = S^1$ with its standard metric, harmonic maps are closed geodesics parametrized proportionally to arc length, or constant maps. So the theorem shows that every free homotopy class of loops in a compact manifold of nonpositive curvature contains a closed geodesic or a constant loop.

The curvature hypothesis cannot be dropped: Eells and Wood showed that, for any metrics, there is no harmonic map of degree $\pm 1$ from a $2$-torus to a $2$-sphere.

Going deeper

For a student

This was the first general existence theorem for harmonic maps, and it introduced the heat flow method into geometry. Hartman later showed that the whole flow converges, not only a sequence, and proved uniqueness results for harmonic maps in a homotopy class under negative curvature. Without the curvature condition, the flow can form singularities in finite or infinite time, and two-dimensional domains show bubbling of harmonic spheres. Harmonic maps from this theorem are a basic tool in rigidity theory, for example in proofs of Mostow-type rigidity and in non-abelian Hodge theory.