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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