Harmonic
Functions |

**In General**

$ __Def__: A function *f* is
harmonic if it satisfies ∇^{2}*f* =
0, the Laplace equation with respect to some Riemannian metric.

* __Conjugate harmonic functions__:
The harmonic conjugate of a function *u*(*x*, *y*)
is another function *v*(*x*, *y*) such that *f*(*x*,* y*)
= *u*(*x*,* y*) + i *v*(*x*,* y*)
satisfies
the Cauchy-Riemann conditions; Given by *v*(*z*) = ∫ (*u*_{x}
d*y* – *u*_{y}
d*x*).

@ __On manifolds__: Colding & Minicozzi AM(97).

**Results and Applications** > s.a. Earnshaw's
Theorem.

* __In general__: A harmonic
function can have no maxima or minima.

* __On a non-compact manifold__:
A bounded harmonic function on a non-compact manifold is constant; A positive
harmonic function on a non-compact manifold with non-negative *R* is constant (& Yau).

* __In Euclidean space__: The average of *f* over a sphere is equal to
its value at the center.

**Harmonic Coordinates** > s.a. coordinates / D'Alembertian; gauge choice.

* __History__: Introduced by Lanczos, DeDonder and Georges Darmois for the Einstein equation, which in vacuum then looks like a quasidiagonal, quasilinear system of second-order partial differential equations hyperbolic for a Lorentzian metric; Initially called "isotherm".

* __Idea__: A choice of gauge
(a.k.a. Lorenz gauge) for generally covariant theories.

* __In general__: Defined
by ∂_{a}(|*g*|^{1/2} *g*^{ab})
= 0, or equivalently
∇^{2}*x*^{a} =
0; Thus, they always give Γ^{k}
:= *g*^{ij} Γ^{k}_{ij}
= 0.

* __For 2D manifolds__: It
is particularly useful to use harmonic functions as coordinates; Given a harmonic
function *α* there exists always (locally) a conjugate harmonic function *β*,
such that *g*_{ab} is explicitly
written in the conformally flat form [@ Wald 84, problem 3.2, p53].

* __For flat space__: They are actually harmonic functions of the Cartesian
coordinates.

* __Applications__: Harmonic coordinates are used in applied relativity as a practical tool for calculating motion of celestial bodies (planets, Moon, satellites); The International Astronomical Union has adopted these coordinates as the basis for doing numerical ephemerides and time metrology in the solar system.

@ __General references__: in Weinberg 72; Bičák & Katz CzJP(05)gq [stationary
asymptotically flat, with matter].

@ __In relativistic celestial mechanics__: Kopeikin CelM(88), Brumberg & Kopeikin NCB(89) [in curved spacetime]; Soffel et al AJ(03)ap [adoption by the IAU]; Kopeikin et al 11; Soffel & Langhans 13.

**Harmonic Analysis**

@ __References__: Stein ed-84; Deitmar 05 [II, primer].

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