Harmonic map

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, N is made of marble (their shapes given by their respective metrics), and 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.

Mathematical definition

Given M, N and φ as above, and denoting by g and h the metrics on M and N, the energy of φ at a point $x\in M$ is defined as $e(\varphi )(x)={\frac {1}{2}}{\rm {trace}}_{g}\varphi ^{*}h$.

In local coordinates, the right hand side of this equality reads ${\frac {1}{2}}g^{ij}h_{\alpha \beta }{\frac {\partial \varphi ^{\alpha }}{\partial x^{i}}}{\frac {\partial \varphi ^{\beta }}{\partial x^{j}}}$.

If M is compact, define the total energy of the map φ as $E(\varphi )=\int _{M}e(\varphi )dv_{g}$ (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 this energy functional. 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 ${\rm {trace}}_{g}\nabla d\varphi ^{*}h=0$, where $\nabla$ is the connection on the vector bundle $T^{*}M\otimes \varphi ^{-1}(TN)$ induced by the Levi-Civita connections on M and N.

Examples

• When the source manifold M is R or S¹, 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.)
• When the target manifold N is Rⁿ (with its standard Euclidean metric), 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, $\nabla d\varphi ^{*}h$ itself vanishes, not just its trace).
• Every holomorphic map between Kähler manifolds is harmonic.

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.

External links

• MathWorld: Harmonic map
• Harmonic maps bibliography