You need to know: Euclidean space , halfspace of , topological space, (smooth) surface, boundary of a surface, area of a surface, compact surface, connected surface, homeomorphic surfaces, geodesic on a surface (a locally length-minimising curve).
Background: A surface M is called embedded in if it can be placed in without self-intersections. A surface is called a minimal surface if every point has a neighbourhood S, with boundary B, such that S has a minimal area out of all surfaces S’ with the same boundary B. A surface S is called complete if any parametrized geodesic of S, may be extended into a parametrized geodesic defined on the entire line . A surface is said to have finite topology if it is homeomorphic to a compact surface with a finite number of points removed.
The Theorem: On 9th April 2004, Tobias Colding and William Minicozzi II submitted to arxiv and the Annals of Mathematics a paper in which they proved that the plane is the only complete embedded minimal surface with finite topology in a halfspace of .
Short context: Minimal surfaces are central objects of study in low-dimensional topology. In 1965, Calabi conjectured that (i) a complete minimal surface in must be unbounded, and, moreover, (ii) it has an unbounded projection on every line, unless it is a plane. In this generality, this is false: a counterexample to part (ii) is given by Jorge and Xavier in 1980, and to part (i) by Nadirashvili in 1996. However, these examples are not embedded (have self-intersections), and Yau asked in 2000 whether Calabi conjectures are true for embedded surfaces. Because every surface which cannot be covered by any halfspace is unbounded, and has an unbounded projection on every line, the Theorem confirms both conjectures for embedded surfaces with finite topology.
Links: Free arxiv version of the original paper is here, journal version is here. See also Section 8.1 of this book for an accessible description of the Theorem.
Theorems of the 21st century 