Harmonic Functions  

In General
$ Def: A function f is harmonic if it satisfies ∇2f = 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) = (ux dyuy dx).
@ 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 gab) = 0, or equivalently ∇2xa = 0; Thus, they always give Γk := gij Γkij = 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 gab 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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