In 1982, Celso José da Costa wrote down the equations of a minimal surface that most mathematicians at that time thought shouldn’t exist. It was a complete minimal torus with two catenoidal and one planar end. From far away, it looks like a catenoid intersected by a horizontal plane.
C. Costa: Imersöes minimas en R³ de gênero un e curvatura total finita, PhD thesis IMPA, Rio de Janeiro, Brasil 1982.
D. Hoffman and W. Meeks, A Complete Embedded Minimal Surface in R³ with Genus One and Three Ends, Journal of Differential Geometry 21, 109-127 (1985)