Harmonic
Maps |

**In General** > s.a. Totally
Geodesic Mapping.

$ __Def__: A map *f *: *M* → *N* between
two manifolds with metric, (*M*,* g*_{ab})
and (*N*, *h*_{AB}),
with coordinates respectively {*x*^{a}}
and {*φ*^{A}},
is said to be a harmonic map if it extremizes the energy functional

*E*[*f*]:= \(1\over2\)∫_{M} *h*_{AB} (∇_{a}
*φ*^{A})
(∇_{b} *φ*^{B})
*g*^{ab} |*g*|^{1/2} d*x* , i.e.
, δ*E*[*f*] = 0 .

* __Terminology__: (*M*,* g*)
is called the base space and (*N*,* h*) the target space.

* __Euler-Lagrange equations__:

*φ*^{A}_{,a}^{;a}
= 0 , where *φ*^{A}_{;ab}:=
∂_{b} *φ*^{A}_{,a}– *φ*^{A}_{c} Γ^{c}_{ab}
+ Γ^{A}_{BC} *φ*^{B}_{,a} *φ*^{C}_{,b} ,

or |*g*|^{–1/2} ∂_{a}(|*g*|^{1/2}g^{ab} ∂_{b}*φ*^{A})
+ *g*^{ab} Γ^{A}_{BC} ∂_{a}*φ*^{B} ∂_{b}*φ*^{C} =
0 .

* __Properties__: *E*[*f*]
is positive definite if (*M*, *g*) and (*N*, *h*) are Riemannian;
The equations δ*E*[*f*] = 0 are elliptic iff (*M*,* g*) is Riemannian.

**References**

@ __General__: Fuller PNAS(54) [introduced them and terminology]; Eells & Samson AJM(64).

@ __Reviews__: Eells & Lemaire BLMS(78); Eells & Lemaire 83.

@ __Related topics__: Hardt BAMS(97)
[singularities].

**Examples**

* __1D target space__: The
simplest example; We get the Laplace equation or the wave equation (depending
on the signature of *g*), which defines the harmonic functions,

∂_{m}(|*g*|^{1/2}*g*^{mn}
∂_{n}*φ*) = 0 .

* __1D base space__: If *M* = E^{1} (1D Euclidean space), we get
the action for geodesics and the geodesic equation.

* __2D base space__: If *M* = \(\mathbb R\)^{2}, we get the membrane problem
(if...).

* __Other special types__:

- Any isometry, covering or minimal immersion of Riemannian manifolds *M* →
*M*'.

- Any homomorphism of compact semisimple Lie groups *G* → *G*'.

- Any holomorphic map of Kähler manifolds.

@ __References__: Bizoń PRS(95)
[S^{3} → S^{3}];
Bizoń & Chmaj PRS(97),
Chiakuei & Zizhou
Top(98)
[spheres]; Speight m.DG/01 [L^{2} metric
on harmonic maps S^{2 }→ S^{2} or \(\mathbb R\)P^{2 }→ \(\mathbb R\)P^{2}];
Daskalopoulos & Mese JDG(08) [from a simplicial complex].

**Applications in Physics** > s.a. sigma
models; embeddings.

* __Common cases__: Usually
(*M*,* g*) is flat and (*N*,* h*)
is the set of values of some naturally non-linear field.

* __In general relativity__:
Harmonic maps have been used extensively in general relativity, e.g., to find
families of solutions of the Einstein equation (> see Ernst
Equation) or Einstein-Yang-Mills theory, or in connection with the black-hole uniqueness
theorems, using in all these cases the existence of one or two Killing vector fields.

* __In particle physics__: They have been used in *σ*-models.

@ __References__: Misner PRD(78);
Guest 97 [loop groups and integrable systems]; Nutku in(93)gq/98 [colliding
electrovacuum waves]; Corlette & Wald CMP(01)mp/99; Ren & Duan a1703 [connecting general relativity with classic chaos and quantum theory].

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