Harmonic map

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A (smooth) map φ:M→N between Riemannian manifolds M and N is called harmonic if it is a critical point of the energy functional E(φ).

This functional E will be defined precisely below—one way of understanding it is to imagine that M is made of rubber and N made of marble (their shapes given by their respective metrics), and that the map φ:M→N prescribes how one "applies" the rubber onto the marble: E(φ) then represents the total amount of elastic potential energy resulting from tension in the rubber. In these terms, φ is a harmonic map if the rubber, when "released" but still constrained to stay everywhere in contact with the marble, already finds itself in a position of equilibrium and therefore does not "snap" into a different shape.

Harmonic maps were introduced in 1964 by J. Eells and J.H. Sampson.^[1]^[2]^[3]

1 Mathematical definition
2 Examples
3 Problems and applications
4 References
5 External links

[edit] Mathematical definition

Given M, N and φ as above, denote by g and h the metrics on M and N. Then the energy of φ at a point x in M is defined as e(φ)(x)= $\textstyle\frac12$ trace_gφ^*h.

In local coordinates, the right hand side of this equality reads $\textstyle\frac12g^{ij}h_{\alpha\beta}\frac{\partial\varphi^\alpha}{\partial x^i}\frac{\partial\varphi^\beta}{\partial x^j}$ .

If M is compact, define the total energy of the map φ as E(φ)= $\textstyle\int$ _Me(φ)dv_g (where dv_g denotes the measure on M induced by its metric).

Then φ is called a harmonic map if it is an critical point of the energy functional E. This definition is extended to the case where M is not compact by asking the restriction of φ to every compact domain to be harmonic.

Alternatively, the map φ is harmonic if it satisfies the Euler-Lagrange equations associated to the functional E. These equations read trace_g∇dφ^*h=0, where ∇ is the connection on the vector bundle T^*M⊗φ^-1(TN) induced by the Levi-Civita connections on M and N.

[edit] Examples

Assume that the source manifold M is the real line R (or the circle S¹), i.e. that φ is a curve (or a closed curve) on N. Then φ is a harmonic map if and only if it is a geodesic. (In this case, the rubber-and-marble analogy described above reduces to the usual elastic band analogy for geodesics.)
Assume that the target manifold N is Euclidean space Rⁿ (with its standard metric). Then φ is a harmonic map if and only if it is a harmonic function in the usual sense (i.e. a solution of the Laplace equation). This follows from the Dirichlet principle.
Every minimal immersion is a harmonic map.
Every totally geodesic map is harmonic (in this case, ∇dφ^*h itself vanishes, not just its trace).
Every holomorphic map between Kähler manifolds is harmonic.

[edit] Problems and applications

If, after applying the rubber M onto the marble N via some map φ, one "releases" it, it will try to "snap" into a position of least tension. This "physical" observation leads to the following mathematical problem: given a homotopy class of maps from M to N, does it contain a representative that is a harmonic map?
Existence results on harmonic maps between manifolds has consequences for their curvature.
Once existence is known, how can a harmonic map be constructed explicitly? (One fruitful method uses twistor theory.)
In theoretical physics, harmonic maps are also known as sigma models.

[edit] References

^ J. Eells and J.H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160
^ J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1–68
^ J. Eells and L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), 385–524