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.
This surface was proven to be embedded in 1985 by David Hoffman and William Meeks. Its discovery has triggered an enormous amount of research, and many open conjectures are tied to it.
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)