You need to know: Surface, orientable surface, connected surface, boundary of a surface, area of a surface, compact surface, homeomorphic surfaces.
Background: 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 is called properly embedded in , if it has no boundary, no self-intersections, and its intersection with any compact subset of is compact. The genus of a connected orientable surface is the maximum number of cuttings along non-intersecting closed simple curves without making it disconnected. A surface is said to have finite topology if it is homeomorphic to a compact surface with a finite number of points removed. The number of points removed is called the number of ends of a surface.
The Theorem: On 20th June 2013, David Hoffman, Martin Traizet, and Brian White submitted to Acta Mathematica a paper in which they proved that for every positive integer g, there exists a connected, properly embedded minimal surface in with one end and genus g.
Short context: Meeks III and Rosenberg proved that the only examples of properly embedded simply connected minimal surfaces in are plane and helicoid. Later, Weber, Hoffman, and Wolf constructed a properly embedded minimal surface in with one end and genus . The Theorem proves that such surface exists for every genus g.
Links: Free arxiv version of the original paper is here, journal version is here.
Theorems of the 21st century 
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