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 9th May 2016, William Meeks III, Joaquin Perez, and Antonio Ros submitted to arxiv and Acta Mathematica a paper in which they proved that for every positive integer g, there exists a constant such that any properly embedded minimal surface in with genus g and finite topology has at most ends.
Short context: Minimal surfaces remain an active area of research since 19th century, and this research area has a golden age at the beginning of the 21st century, with a large number of impressive results, see, for example, here, here, and here. One important conjecture of Hoffman and Meeks states that if a properly embedded connected minimal surface of genus g has a finite number k ends, then . However, before 2016, it was not known if there exists any upper bound on k depending only on g. This is what the Theorem establishes.
Links: Free arxiv version of the original paper is here, journal version is here.
Theorems of the 21st century 