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 \(\delta E[f] = 0\) 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}
∂_{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 \({\rm S}^2 \to {\rm S}^2\) 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].

main page
– abbreviations
– journals – comments
– other sites – acknowledgements

send feedback and suggestions to bombelli at olemiss.edu – modified 26 oct 2018