Harmonic
Maps| Harmonic
Maps |
In General > s.a. Totally
Geodesic Mapping.
$ Def: A map f : M → N between
two manifolds with metric, (M, gab)
and (N, hAB),
with coordinates respectively {xa}
and {
A},
is said to be a harmonic map if it extremizes the energy functional
E[f]:= 
M hAB (
a
A)
(b B)
gab |g|1/2 dx , i.e.
, 
E[f]
= 0 .
* Terminology: (M, g)
is called the base space and (N, h)
the target space.
* Euler-Lagrange equations:
A,a;a
= 0 , where A;ab:=
b A,a– Ac 
cab
+ ABC B,a C,b ,
or |g|–1/2 a(|g|1/2gab bA)
+ gab ABC aB bC =
0 .
* Properties: E[f]
is positive definite if (M, g)
and (N, h) are Riemannian;
The equations 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|1/2gmn
n)
= 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 = R2, 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: Bizon PRS(95)
[S3 → S3];
Bizon & Chmaj PRS(97),
Chiakuei & Zizhou
Top(98)
[spheres]; Speight m.DG/01 [L2 metric
on harmonic maps S2 → S2 or RP2 → RP2];
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 
-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.
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send feedback and suggestions to bombelli at olemiss.edu – modified 21
jul
2009