# Harmonic map

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{{#invoke:Hatnote|hatnote}} A (smooth) map φ:MN between Riemannian manifolds M and N is called harmonic if it is a critical point of the Dirichlet energy functional

$E(\phi )=\int _{M}\|d\phi \|^{2}\,d\operatorname {Vol} .$ 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:MN 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 are the 'least expanding' maps in orthogonal directions.

Existence of harmonic maps from a complete Riemannian manifold to a complete Riemannian manifold of non-positive sectional curvature was proved by Template:Harvtxt.

## Mathematical definition

Given Riemannian manifolds (M,g), (N,h) and φ as above, the energy density of φ at a point x in M is defined as

$e(\phi )={\frac {1}{2}}\|d\phi \|^{2}$ where the $\|d\phi \|^{2}$ is the squared norm of the differential of $\phi$ , with respect to the induced metric on the bundle $T^{*}M\otimes \phi ^{-1}TM$ . The total energy of φ is given by integrating the density over M

$E(\phi )=\int _{M}e(\phi )\,dv_{g}={\frac {1}{2}}\int _{M}\|d\phi \|^{2}\,dv_{g}$ where dvg denotes the measure on M induced by its metric. This generalizes the classical Dirichlet energy.

The energy density can be written more explicitly as

$e(\phi )={\frac {1}{2}}\operatorname {trace} _{g}\phi ^{*}h.$ Using the Einstein summation convention, in local coordinates the right hand side of this equality reads

$e(\phi )={\frac {1}{2}}g^{ij}h_{\alpha \beta }{\frac {\partial \phi ^{\alpha }}{\partial x^{i}}}{\frac {\partial \phi ^{\beta }}{\partial x^{j}}}.$ If M is compact, then φ is called a harmonic map if it is a critical point of the energy functional E. This definition is extended to the case where M is not compact by requiring the restriction of φ to every compact domain to be harmonic, or, more typically, requiring that φ be a critical point of the energy functional in the Sobolev space H1,2(M,N).

Equivalently, the map φ is harmonic if it satisfies the Euler-Lagrange equations associated to the functional E. These equations read

$\tau (\phi )\ {\stackrel {\text{def}}{=}}\ \operatorname {trace} _{g}\nabla d\phi =0$ where ∇ is the connection on the vector bundle T*M⊗φ−1(TN) induced by the Levi-Civita connections on M and N. The quantity τ(φ) is a section of the bundle φ−1(TN) known as the tension field of φ. In terms of the physical analogy, it corresponds to the direction in which the "rubber" manifold M will tend to move in N in seeking the energy-minimizing configuration.

## 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, a quantum field theory whose action is given by the Dirichlet energy is known as a sigma model. In such a theory, harmonic maps correspond to instantons.
• One of the original ideas in grid generation methods for computational fluid dynamics and computational physics was to use either conformal or harmonic mapping to generate regular grids.

## Harmonic maps between metric spaces

The energy integral can be formulated in a weaker setting for functions u : MN between two metric spaces Template:Harv. The energy integrand is instead a function of the form

$e_{\epsilon }(u)(x)={\frac {\int _{M}d^{2}(u(x),u(y))\,d\mu _{x}^{\epsilon }(y)}{\int _{M}d^{2}(x,y)\,d\mu _{x}^{\epsilon }(y)}}$ in which μTemplate:Su is a family of measures attached to each point of M.