On the minimality condition for caustics of pseudo-spherical surfaces (Q6554705)

The paper under review deals with elementary differential geometric properties of pseudo-spherical surfaces in \(\mathbb R^3\). Given an arbitrary pseudo-spherical surface \(F\), which is viewed rather as a front, the authors consider two caustics associated with \(F\) and prove that either of these caustics is a minimal surface if and only if \(F\) belongs to the family of Dini surfaces corresponding to the one-soliton solution of the sine-Gordon equation, c.f. [\textit{K. Teramoto}, Forum Math. 36, 21--32 (2024; Zbl 1536.53018)]. Moreover, it is shown that caustics of Dini surfaces belong to the associated family of catenoids and helicoids.