# Minimal surfaces

Among the definitions of minimal suraface I found these two:

(1) A surface $$M\subset\mathbb{R}^3$$ is minimal if for any point $$p\in M$$ there is a neighborhood $$U$$ of $$p$$ in $$M$$ that minimizes the area relatively to its boundary.

(2) A surface $$M\subset\mathbb{R}^3$$ with zero mean curvature.

I would like to understand why (1) and (2) are equivalent. Thank you very much.

b. You may allow your minimal surface to have interior branch points. Such surfaces (in $$E^3$$!) are not local area-minimizers (at the branch-points). See for instance,