# Harmonic map

A (smooth) map ${\displaystyle \phi }$:MN between Riemannian manifolds M and N is called harmonic if it is a critical point of the Dirichlet energy functional

${\displaystyle E(\phi )=\int _{M}\|d\phi \|^{2}\,d\operatorname {Vol} .}$

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 ${\displaystyle \phi }$:MN prescribes how one "applies" the rubber onto the marble: E(${\displaystyle \phi }$) then represents the total amount of elastic potential energy resulting from tension in the rubber. In these terms, ${\displaystyle \phi }$ 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 ${\displaystyle \phi }$  as above, the energy density of ${\displaystyle \phi }$  at a point x in M is defined as

${\displaystyle e(\phi )={\frac {1}{2}}\|d\phi \|^{2}}$

where the ${\displaystyle \|d\phi \|^{2}}$  is the squared norm of the differential of ${\displaystyle \phi }$ , with respect to the induced metric on the bundle ${\displaystyle T^{*}M\otimes \phi ^{*}(TN)}$ where ${\displaystyle \phi ^{*}}$ is the pullback. The total energy of ${\displaystyle \phi }$  is given by integrating the density over M

${\displaystyle E(\phi )=\int _{M}e(\phi )\,dv_{g}={\frac {1}{2}}\int _{M}\|d\phi \|^{2}\,dv_{g}}$

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

${\displaystyle e(\phi )={\frac {1}{2}}\operatorname {trace} _{g}\phi ^{*}h.}$

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

${\displaystyle e(\phi )={\frac {1}{2}}g^{ij}h_{\alpha \beta }{\frac {\partial \phi ^{\alpha }}{\partial x^{i}}}{\frac {\partial \phi ^{\beta }}{\partial x^{j}}}.}$

If M is compact, then ${\displaystyle \phi }$  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 ${\displaystyle \phi }$  to every compact domain to be harmonic, or, more typically, requiring that ${\displaystyle \phi }$  be a critical point of the energy functional in the Sobolev space H1,2(M,N).

Equivalently, the map ${\displaystyle \phi }$  is harmonic if it satisfies the Euler-Lagrange equations associated to the functional E. These equations read

${\displaystyle \tau (\phi )\ {\stackrel {\text{def}}{=}}\ \operatorname {trace} _{g}\nabla d\phi =0}$

where ∇ is the connection on the vector bundle ${\displaystyle T^{*}M\otimes \phi ^{*}(TN)}$ induced by the Levi-Civita connections on M and N. The quantity τ(${\displaystyle \phi }$ ) is a section of the bundle ${\displaystyle \phi }$ *(TN) known as the tension field of ${\displaystyle \phi }$ . 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.

## Examples

• Identity and constant maps are harmonic.
• Assume that the source manifold M is the real line R (or the circle S1), i.e. that ${\displaystyle \phi }$  is a curve (or a closed curve) on N. Then ${\displaystyle \phi }$  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 ${\displaystyle \phi }$  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. If ${\displaystyle \phi }$  is a diffeomorphism onto an open set in Rn, then it gives a harmonic coordinate system.
• Every minimal surface in Euclidean space is a harmonic immersion.
• More generally, a minimal submanifold M of N is a harmonic immersion of M in N.
• Every totally geodesic map is harmonic (in this case, ∇d${\displaystyle \phi }$ *h itself vanishes, not just its trace).
• Every holomorphic map between Kähler manifolds is harmonic.
• Every harmonic morphism between Riemannian manifolds is harmonic.

## Problems and applications

• If, after applying the rubber M onto the marble N via some map ${\displaystyle \phi }$ , 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

${\displaystyle e_{\epsilon }(u)(x)={\frac {\int _{M}d^{2}(u(x),u(y))\,d\mu _{x}^{\epsilon }(y)}{\int _{M}d^{2}(x,y)\,d\mu _{x}^{\epsilon }(y)}}}$

in which με
x
is a family of measures attached to each point of M. 45613213212

## References

• Eells, J.; Sampson, J.H. (1964), "Harmonic mappings of Riemannian manifolds", Amer. J. Math., 86: 109–160, doi:10.2307/2373037, 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.