Curvature Flow and Green's Functions

Question

I'm trying to find out if there is a research area with associated literature linking the solution of Laplace's and Helmholtz's equations in 3D to curvature flow. If you look at the solution of Laplace's and Helmholtz's equations for a cylinder or sphere it appears there is a relationship with mean curvature and maybe Gaussian curvature and the Taylor expansion of the Green's functions for a line and point source respectively. This has potential applications in computational geometry. For Helmholtz, the sphere/point Green's function is $\frac{e^{jk |x-x_0| }}{4 \pi |x-x_0|}$ and the cylinder/line Green's function is $\approx C\frac{e^{jk |x-x_0| }}{\sqrt{|x-x_0|}}$. The "damped" versions are: $\frac{e^{-\beta |x-x_0| }}{4 \pi |x-x_0|}$ and $\approx C\frac{e^{-\beta |x-x_0| }}{\sqrt{|x-x_0|}}$, respectively. It appears the denominator encodes the curvature information. Any input is appreciated.