Global analysis, analysis on manifolds Harmonic maps and gauge-theoretic variational problems Theorem

Hartman's uniqueness theorem for harmonic maps

Builds on Harmonic map Riemannian manifold Sectional curvature Homotopy Geodesic Energy of a map between Riemannian manifolds Eells–Sampson theorem

Statement

Let $M$ be a compact Riemannian manifold without boundary. Let $N$ be a compact Riemannian manifold whose sectional curvature is nonpositive everywhere.

Let $u_0, u_1 \colon M \to N$ be harmonic maps that are homotopic. Then 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 as $u_0$ . Moreover, for each $x \in M$ the curve $t \mapsto u_t(x)$ is a geodesic traversed at constant speed, and this speed does not depend on $x$ .
Suppose that the sectional curvature of $N$ is negative everywhere. Let $u \colon M \to N$ be a harmonic map whose image is neither a point nor a closed geodesic. Then $u$ is the only harmonic map in its homotopy class.

Remarks

Lift 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. In nonpositive curvature the squared distance between the two lifted maps is a subharmonic function on $M$ , so it is constant on the compact manifold $M$ . Following the geodesics gives the homotopy $u_t$ , and the second variation of the energy shows that every $u_t$ is again harmonic. With 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.

The curvature hypothesis is needed. For $M = N = S^2$ with the round metric, every rotation is a harmonic map homotopic to the identity. So harmonic maps in a homotopy class are far from unique. Uniqueness 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.

Going deeper

For a student

The theorem complements the Eells–Sampson theorem, which gives existence of a harmonic map in each homotopy class under the same curvature hypothesis. Together they say that, for targets of negative curvature, a homotopy class of maps usually has a canonical harmonic representative. This is used in rigidity theory, for example for maps between locally symmetric spaces and in the study of Teichmüller space.

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Energy of a map between Riemannian manifolds concept
Geodesic concept
Harmonic map concept
Harmonic map heat flow concept
Homotopy concept
Riemannian manifold concept
Sectional curvature concept
Eells–Sampson theorem statement
Hartman's uniqueness theorem for harmonic maps statement

Tag	`3VMJGQCD7S`
URI	`m2://3VMJGQCD7S`
Kind	statement, theorem
Writing	W0 stub
MSC	58E20, 53C43
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About	Harmonic map
Uses	Harmonic map, Riemannian manifold, Sectional curvature, Homotopy, Geodesic, Energy of a map between Riemannian manifolds, Eells–Sampson theorem