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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.

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It depends on what you mean by a surface:

a. If your surface is smooth then the two definitions agree. This can be seen as follows. Every smooth surface is locally a graph. For graphs, minimality (in the sense of zero mean curvature which is the standard definition) is equivalent to the property that the surface is an area-minimizer.

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,

Mario J. Micallef and Brian White, The Structure of Branch Points in Minimal Surfaces and in Pseudoholomorphic Curves, Annals of Mathematics, vol. 141, no. 1, 1995, pp. 35–85.

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