Question on the relationship between the surface bounded by Jordan curve and critical point I was reading 'Notes on Minimal surface' by Michael Beeson. I am confused on top of page 11 that saying   'A surface of least area bounded by Γ would be a critical point of A, but not necessarily conversely'. Can anyone explain or prove the statement and give an example showing the converse is false. Based on my understanding, in this book a Jordan curve is defined to be a continuous injective map that sends a unit circle to R^3. Would a surface bounded by the Jordan curve be part of the sphere? For the counterexample, I was thinking is there exists a boundary that bounded more than one minimal surfaces?
Many thanks. The book can be found here
 A: Assuming $A$ (which you do not introduce or define) is continuous with respect to deformation of the surface, then the concepts in play are elementary optimization, usually taught in Calculus I.  In this context, a critical point is a point where the derivative of $A$ with respect to deformation of the surface is either zero or undefined.
A differentiable local minimum of $A$ (simple model, $x^2$ at $x = 0$), some differentiable inflection points of $A$ (simple model, $x^3$ at $x = 0$), a differentiable local maximum of $A$ (simple model, $-x^2$ at $x = 0$), and points where the derivative of $A$ is undefined (simple model, $|x|$ at $x = 0$) are critical points.  Since a global minimum of a continuous function is either a differentiable local minimum or is a nondifferentiable point, any global minimum is a critical point.
Of course, excluding that $A$ is constant under all deformations of the surface, other local minima,  local maxima, other stationary points (inflections in the previous list), and other points of nondifferentiability are also critical points, but none of these are the global minimum.
It can happen that more than one point in the domain of $A$ gives the same value as the global minimum of $A$ (simple model, $-\cos x$ at $x = 2 \pi k$ for any integer $k$).  It is reported that Enneper's wire is an example of such a curve.  It is know that a (reasonable) curve cannot be the boundary of infinitely many different minimal surfaces.  (So,the real behaviour of the minima of $A$ can't quite replicate the infinitely many global minima of the simple model $-\cos x$.)  So each (reasonable) curve bounds at least one and at most some finite number of minimal surfaces, all having the same area.
