Let $f(z)dz^2$ be a holomorphic quadratic differential on the punctured disk $\{0<|z|<1\}$, which gives rise to a Riemannian metric $g=|f(z)|\,|dz|^2$ and hence a volume form $\nu=|f(z)| dx\,dy$.

Problem:prove that if $f$ has essential singularity at $0$, then(1) the total area of $\nu$ is infinite;

(2) the metric $g$ is incomplete at $0$, i.e. there is a sequence of points $(z_n)_{n\geq 0}$ which converges to $0$ while the distances $d(z_0, z_n)$ (defined with respect to $g$) are bounded.

I don't know much about essential singularities except Picard's theorem, which is not enough for either of the questions. Standard techniques such as Maximum Modulus Principle don't seem enough either.

In Strebel's book Quadratic Differentials, it is claimed (in the paragraph following Definition 5.3) that a neighborhood of $0$ has finite area with respect to $\nu$ if and only if $0$ is a first order pole of $f$. Proposition (1) above is the nontrivial portion of this claim. But I don't know whether Strebel had considered essential singularities seriously since this is discussed nowhere in his book.