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  "body": "\n## Statement\n\nLet $M$ be a compact [[Riemannian manifold|7CBT4SM83Y]] without boundary.\nLet $N$ be a compact Riemannian manifold whose [[sectional curvature|1X42JAA349]] is nonpositive everywhere.\n\n1. Let $u_0, u_1 \\colon M \\to N$ be [[harmonic maps|KZJYG9DS3M]] that are [[homotopic|9YA6S6HD9V]].\nThen there is a smooth homotopy $u_t$, $t \\in [0, 1]$, from $u_0$ to $u_1$ such that each $u_t$ is harmonic and has the same [[energy|E6HDE1DFXD]] as $u_0$.\nMoreover, for each $x \\in M$ the curve $t \\mapsto u_t(x)$ is a [[geodesic|NSYMQMHPDN]] traversed at constant speed, and this speed does not depend on $x$.\n2. Suppose that the sectional curvature of $N$ is negative everywhere.\nLet $u \\colon M \\to N$ be a harmonic map whose image is neither a point nor a closed geodesic.\nThen $u$ is the only harmonic map in its homotopy class.\n\n## Remarks\n\nLift to the universal cover of $N$, which is a Hadamard manifold, and join $u_0(x)$ to $u_1(x)$ by the unique geodesic in the right homotopy class.\nIn nonpositive curvature the squared distance between the two lifted maps is a subharmonic function on $M$, so it is constant on the compact manifold $M$.\nFollowing the geodesics gives the homotopy $u_t$, and the second variation of the energy shows that every $u_t$ is again harmonic.\nWith strictly negative curvature the homotopy must be constant, unless all the geodesics $t \\mapsto u_t(x)$ run along one closed geodesic, which forces the image of $u$ into that geodesic.\n\nThe curvature hypothesis is needed.\nFor $M = N = S^2$ with the round metric, every rotation is a harmonic map homotopic to the identity.\nSo harmonic maps in a homotopy class are far from unique.\nUniqueness also fails in the flat case: on a flat torus $N$, composing a harmonic map with a translation gives a family of homotopic harmonic maps, as part 1 allows.\n\n## Explanations\n\n### L1 human\n\nWhen the target space curves like a saddle everywhere, harmonic maps are rigid.\nTwo harmonic maps that can be deformed into each other differ only by sliding along geodesics in a uniform way.\nWhen the curvature is strictly negative, even this sliding is impossible except in degenerate cases, so the harmonic map is unique.\n\n### L2 human\n\nThe theorem complements the [[Eells–Sampson theorem|AP4A0GX0V2]], which gives existence of a harmonic map in each homotopy class under the same curvature hypothesis.\nTogether they say that, for targets of negative curvature, a homotopy class of maps usually has a canonical harmonic representative.\nThis is used in rigidity theory, for example for maps between locally symmetric spaces and in the study of Teichmüller space.\n",
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