I am reading Lecture Notes on Minimal Surfaces, which can be accessed here and I am trying to find a proof of the following:
Mean curvature of a surface $S$ is zero everywhere if and only if surface $S$ has locally minimal area.
In the section Why a Minimal Surface is Minimal (or Critical) of the mentioned notes, it is shown, that if $H = 0$ everywhere, then area function has a critical point at a chosen point. I wonder what this means; do there exist surfaces, which have mean curvature $H = 0$, but do not have locally minimal area? What is an example of such function? If not, how do we show that this extrema is in fact a local minimum?
It is also shown that if $S$ minimizes area, then its mean curvature vanishes everywhere. According to these two results, the Local least area definition and Mean curvature definition on the wikipedia page are not equivalent; mean curvature implies local least area, but not the other way around. What is happening? What am I understanding wrong?
How can I complete the proof to give me the result I desire (the equivalence of two definitions), does it even hold? Perhaps, you can refer me to a completely unrelated proof of this equivalence, I am having trouble finding it proven anywhere.