Scherk’s fifth minimal surface

Question

Scherk’s fifth minimal surface is defined implicitly by $$ \sin(z)=\sinh(x) \sinh(y). $$ How can I show that this surface is minimal?

Answer 1

From my 1991 article, write this as the zero set of $$ F(x,y,z) = \sinh x \sinh y - \sin z $$ and calculate the mean curvature, which comes out as zero. We have the variables re-named as $x_1, x_2, x_3.$ The shorthand $F_i$ means the partial derivative by $x_i,$ and $F_{ij}$ the second derivative by $x_i$ and $x_j.$ Oh, $\delta_{ij}$ equals the constant $1$ when $i=j,$ otherwise it is the constant $0.$ See http://en.wikipedia.org/wiki/Kronecker_delta