So the equation of a surface with topology S2 can be expanded out in terms of spherical harmonic functions. (I believe). A torus T2 which is just S1xS1 can be expanded out in terms of ordinary harmonics which is just Fourier analysis.
But what about surfaces of higher genus, for example a double torus?
What is the equivalent harmonic functions, or Fourier series for these?
Edit: I am talking about how an arbitrary closed 1D curve can be represented as a Fourier series in terms of sines and cosines (as is done in string theory for example). So in terms of a 2D surface I guess the metric would be defined by its embedding in R3. Wouldn't the series expansion determine the shape of the surface and hence the metric not the other way round?