It can be shown that the Gauss map of a minimal surface is conformal quite easily, but does the converse of this statement hold? Specifically, what surfaces have a conformal Gauss map?
2026-03-26 16:05:38.1774541138
Gauss map of a minimal surface converse
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HINT: The characteristic polynomial of the Weingarten (derivative of Gauss) map at a point $p$ is, of course, $t^2-2H(p)t+K(p)$. Apply the Cayley-Hamilton Theorem. (You might first observe that the Gauss map is always conformal at an umbilic point.)
Even easier, write the Weingarten map in diagonal form. When is it conformal?