Okay, I'm not very comfortable with either complex geometry or minimal surfaces, so bear with me. I've encountered the following theorem, but I can't find a proof:
Suppose $V\subset \mathbb{C}^n$ is a properly embedded complex submanifold, say of complex-dimension $k$. Here, $\mathbb{C}^n$ is equipped with the usual Euclidean metric. The theorem says that for all $p\in V$,
$$\lim_{r\to 0}\frac{\operatorname{Vol}(V\cap B(r,p))}{r^{2k}} > 0.$$
Any thoughts about why this is true, or maybe an idea for a reference? Thanks!