Area of circle in terms of Gaussian curvature

Question

I am asking about a formula in section 2 of these notes.

Let $\rho|dz|$ be a conformal metric on $U\subset\mathbf C$. Then the Gaussian curvature of $\rho|dz|$ at $z\in U$ is defined as $$K_\rho(z)=-\frac{\Delta\log\rho(z)}{\rho^2(z)},$$ where $\Delta$ is the Laplacian.

Then it is claimed that one can show that $K_\rho(z)$ governs the next-to-leading-order term in an asymptotic series development for the area of a circle centered at $z$: $$\operatorname{Area}(B(z,r))=\pi r^2\left(1-\frac1{12}K_\rho(z)r^2\right)+o(r^4).$$

Some references are provided in the notes, but I haven't been able to find a clear proof for this formula (which has the name of Diquet attached to it) in them or elsewhere.

This is a fundamental formula – could you point to me to a clear proof (that is not hideous)? Thanks a bunch.

Answer 1

I found a clean proof in section 2.2, Theorem 2.2.1 of John Hubbard's 'Teichmüller Theory and Applications.'