a regular neighborhood of , is a handlebody as is . Thus the boundary of V in is a Heegaard splitting and this is the standard splitting of . It was proved by Charles Frohman and Joel Hass that any other genus 3 Heegaard splitting of the three-torus is topologically equivalent to this one. Michel Boileau and Jean-Pierre Otal proved that in general any Heegaard splitting of the three-torus is equivalent to the result of stabilizing this example. Theorems Alexander's lemma Up to isotopy, there is a unique (piecewise linear) embedding of the two-sphere into the three-sphere. (In higher dimensions this is known as the Schoenflies theorem. In dimension two this is the Jordan curve theorem.) This may be restated as follows: the genus zero splitting of is unique. Waldhausen's theorem Every splitting of is obtained by stabilizing the unique splitting of genus zero. Suppose now that M is a closed orientable three-manifold. Reidemeister–Singer theorem For any pair of splittings and in M there is a third splitting in M which is a stabilization of both. Haken's lemma Suppose that is an essential two-sphere in M and H is a Heegaard splitting. Then there is an essential two-sphere in M meeting H in a single curve. Classifications There are several classes of three-manifolds where the set of Heegaard splittings is completely known. For example, Waldhausen's Theorem shows that all splittings of are standard. The same holds for lens spaces (as proved by Francis Bonahon and Otal). Splittings of Seifert fiber spaces are more subtle. Here, all splittings may be isotoped to be vertical or horizontal (as proved by Yoav Moriah and Jennifer Schultens). classified splittings of torus bundles (which includes all three-manifolds with Sol geometry). It follows from their work that all torus bundles have a unique splitting of minimal genus. All other splittings of the torus bundle are stabilizations of the minimal genus one. As of 2008, the only hyperbolic three-manifolds whose Heegaard splittings are classified are two-bridge knot complements, in a paper of Tsuyoshi Kobayashi. Applications and connections Minimal surfaces Heegaard splittings appeared in the theory of minimal surfaces first in the work of Blaine Lawson who proved that embedded minimal surfaces in compact manifolds of positive sectional curvature are Heegaard splittings. This result was extended by William Meeks to flat manifolds, except he proves that an embedded minimal surface in a flat three-manifold is either a Heegaard