Conformal metrics on the complex plane

Let $M$ be a Riemannian manifold with metric $g$, recall the Laplace-Beltrami operator of the metric is the elliptic operator defined as

In the previous equation we use the Einstein convention of summing over repeated indexes in covariant and contravariant position. If the manifold is an open region in the complex plane, the Theorem of Liouville asserts that there exists a positive function $\lambda$, such that under a suitable change of coordinates, $g = \lambda^2 dz d\bar{z}$, the curvature of this metric can be computed as

where $\Delta_g$ is the Laplace-Beltrami operator of the metric and $\Delta = 4 \partial_z\bar{\partial_z} = \partial^2_x + \partial^2_y$ is the flat Laplacian.