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  "body": "\n## Statement\n\nLet $(M, g)$ and $(N, h)$ be compact [[Riemannian manifolds|7CBT4SM83Y]] without boundary, and assume that $N$ has nonpositive [[sectional curvature|1X42JAA349]].\nLet $u_0 \\colon M \\to N$ be a smooth map.\nThen the [[harmonic map heat flow|MD99PYBCE6]] with initial map $u_0$ has a smooth solution $u$ on $M \\times [0, \\infty)$.\nThere is a sequence $t_k \\to \\infty$ such that $u(\\cdot, t_k)$ converges smoothly to a [[harmonic map|KZJYG9DS3M]] $u_\\infty \\colon M \\to N$, and $u_\\infty$ is [[homotopic|9YA6S6HD9V]] to $u_0$.\nIn particular, every continuous map from $M$ to $N$ is homotopic to a harmonic map.\n\n## Remarks\n\nThe 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 $M$.\nThe curvature term of $N$ enters with a sign that helps because the sectional curvature of $N$ is nonpositive.\nTogether 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.\nSince $\\int_0^\\infty \\int_M \\abs{\\tau(u)}^2 \\, dV_g \\, dt \\le E(u_0)$, the tension tends to zero along a sequence of times, and compactness gives the harmonic limit.\n\nExample: for $M = S^1$ with its standard metric, harmonic maps are closed geodesics parametrized proportionally to arc length, or constant maps.\nSo 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.\n\nThe 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.\n\n## Explanations\n\n### L1 human\n\nTake a map into a space that is curved like a saddle everywhere.\nThe theorem says that it can be relaxed, by continuously lowering its energy, into a harmonic map, the best possible map in its class.\nThe relaxing process is a heat flow for maps.\nNonpositive curvature of the target prevents the map from tearing or concentrating energy at a point.\n\n### L2 human\n\nThis was the first general existence theorem for harmonic maps, and it introduced the heat flow method into geometry.\nHartman 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.\nWithout the curvature condition, the flow can form singularities in finite or infinite time, and two-dimensional domains show bubbling of harmonic spheres.\nHarmonic 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.\n",
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  "msc": [
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  "name": "Eells–Sampson theorem",
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