Harmonic Functions

In General
$ Def: A function f is harmonic if it satisfies ²f = 0, the Laplace equation wrt 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 dy – u_y 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'Alambertian; gauge.
* 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 ²x^a = 0; Thus, they always give ^k := g^{ij k}_ij = 0.
* For 2D manifolds: It is particularly useful to use harmonic functions as coordinates; Given a harmonic function there exists always (locally) a conjugate harmonic function , 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.
@ References: in Weinberg 72; Bicák & Katz CzJP(05)gq [stationary asymptotically flat, with matter].

Harmonic Analysis
@ References: Stein ed-84; Deitmar 05 [II, primer].

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