Wente Torus

A Wente torus is a member of a classical family of immersed tori in three-dimensional Euclidean space with constant nonzero mean curvature and a family of planar lines of curvature. Wente (1985, 1986, 1987) constructed such surfaces, disproving the conjecture that every closed immersed surface of constant mean curvature in is a sphere. The examples necessarily have self-intersections. They do not contradict Alexandrov's theorem that a connected compact embedded surface without boundary and with constant mean curvature in is a sphere.

Abresch (1987) and Walter (1987) gave explicit constructions using elliptic functions. This family can be indexed by reduced rational numbers with . The figure illustrates the three-lobed and five-lobed examples with and , respectively. The mean curvature can be normalized to have magnitude by a homothecy.

For an oriented immersion, its constant value is characterized by

where , , and are the first fundamental form coefficients and , , and are the second fundamental form coefficients. Unlike a minimal surface, a Wente torus has .