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I was trying to understand Nitsche's Elementary Proof of Bernstein's Theorem on Minimal Surfaces (referenced on this question) but on the very first page, the statement reads:

Let the function $z = z(x,y)$ be twice continuously differentiable and satisfy the equation $rt-s^2 = 1$, $r > 0$ for all values of $x$ and $y$. Then $z(x,y)$ is a quadratic polynomial.

I have not understood who $r$, $t$ and $s$ are. At first I thought that $z = r + is$ and $t$ is some real number but then the functions $p,q$ appear and it would make so much more sense that $z = p + iq$. Any help or insight is appreciated.

D. Brito
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  • $x, y, z$ are real values. The graph of $(x, y, z(x,y))$ is a surface. I'm pretty sure that $r, s, t$ are various functions definable from the surface, for instance curvature and torsion at each point. However, I do not recognize the symbolism, so I cannot tell you which functions they are. – Paul Sinclair Jul 24 '19 at 02:30

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Upon looking in further into reference [2] in the article I found that \begin{align} r = z_{xx}\\ s = z_{xy}\\ t = z_{yy} \end{align} And so, for $z$ to be a solution to equation posted in the question it means that $$ z_{xx}z_{yy} - z_{xy}^2 = 1 $$

D. Brito
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