Math Duo Maps the Infinite Terrain of Minimal Surfaces

But mathematicians and scientists often have occasion to consider other types of worlds than the infinite three-dimensional space we’re used to thinking about — worlds that are curved or finite in size, such as the three-dimensional analogues of a sphere or doughnut surface. When we’re looking for the minimal surfaces within a shape, we can consider a new world that consists of all possible finite surfaces that live in the shape — let’s call this world “surface space.” And about six decades later, in a tour-de-force extension of Birkhoff’s ideas, Almgren and Pitts mapped out the topography of surface space for all finite shapes of dimensions three through seven, and then used that topography to prove that such shapes always have at least one closed minimal surface.

Source: www.quantamagazine.org