# Harmonic map

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 Eells & Sampson (1964).

## 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

$e(\phi) = \frac12\|d\phi\|^2$

where the $\|d\phi\|^2$ is the squared norm of the differential of $\phi$, with respect to the induced metric on the bundle $T^*M \otimes \phi^{-1} T M$. 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

$e(\phi) = \frac12\operatorname{trace}_g\phi^*h.$

Using the Einstein summation convention, in local coordinates the right hand side of this equality reads

$e(\phi) = \frac12g^{ij}h_{\alpha\beta}\frac{\partial\phi^\alpha}{\partial x^i}\frac{\partial\phi^\beta}{\partial x^j}.$

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 (Jost 1995). The energy integrand is instead a function of the form

$e_\epsilon(u)(x) = \frac{\int_M d^2(u(x),u(y))\,d\mu^\epsilon_x(y)}{\int_M d^2(x,y)\,d\mu^\epsilon_x(y)}$

in which με
x
is a family of measures attached to each point of M.

## References

• Eells, J.; Sampson, J.H. (1964), "Harmonic mappings of Riemannian manifolds", Amer. J. Math. 86: 109–160, JSTOR 2373037.
• Eells, J.; Lemaire, J. (1978), "A report on harmonic maps", Bull. London Math. Soc. 10: 1–68, doi:10.1112/blms/10.1.1.
• Eells, J.; Lemaire, J. (1988), "Another report on harmonic maps", Bull. London Math. Soc. 20: 385–524, doi:10.1112/blms/20.5.385.
• Jost, Jürgen (1994), "Equilibrium maps between metric spaces", Calculus of Variations and Partial Differential Equations 2 (2): 173–204, doi:10.1007/BF01191341, ISSN 0944-2669, MR 1385525.
• Jost, Jürgen (2005), Riemannian Geometry and Geometric Analysis (4th ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-25907-7.