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 dx ,   i.e. ,   δE[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|^1/2g^ab ∂_bφ^A) + g^ab Γ^A_BC ∂_aφ^B ∂_bφ^C = 0 .

* Properties: E[f] is positive definite if (M, g) and (N, h) are Riemannian; The equations \(\delta E[f] = 0\) 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 ∂_nφ) = 0 .

* 1D base space: If M = E¹ (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) [S³ → S³]; Bizoń & Chmaj PRS(97), Chiakuei & Zizhou Top(98) [spheres]; Speight m.DG/01 [L\(^2\) metric on harmonic maps \({\rm S}^2 \to {\rm S}^2\) or \(\mathbb R{\rm P}^2 \to \mathbb R{\rm P}^2\)]; 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 CSF(17)-a1703 [connecting general relativity with classic chaos and quantum theory].

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