A surface which is simultaneously complete and minimal. There have been a large number of fundamental
breakthroughs in the study of such surfaces in recent years, and they remain the
focus of intensive current research.

Until the Costa minimal surface was discovered in 1982, the only other known complete minimal embeddable surfaces in with no self-intersections were the plane,
catenoid, and helicoid.
The plane is genus 0 and the catenoid and the helicoid are genus 0 with two punctures,
but the Costa minimal surface is genus 1 with three punctures (Schwalbe and Wagon
1999).