We get negative Gauss curvature K surfaces of quasi polar symmetry surface form generated from:

$$Re (x+ i y)^n = a^n$$ (n integer) with n humps above plane $z =0$.

($n =2,3,4$ hyperbolic paraboloids, monkey saddles, four humped frill/ pleated shells respectively).

"Quasi" due to $n>2$ introducing circumferential pleats or frills.

What polynomial or differential relation function yields constant negative surfaces of $K = -1/a^2$?

How to set up its differential equation?

Instead of $z^n$ if a simpler $\cos z$ is taken, no clue or handle how to modify the function in order to get a constant negative$K. ( \cos x \cosh y =$ const)

• Embeddings of portions of the hyperbolic plane into Euclidean 3-space has been classically studied. mathoverflow.net/questions/533/… gives some references. – Willie Wong Oct 26 '15 at 20:14
• There is no unique mapping it appears. About the Coral ruffles (in comments of Oct 15 above) ,I sent a message to Prof. Daina Taimina sometime back, says there is no known parameterization. – Narasimham Oct 26 '15 at 21:12