A constant mean curvature surface is the image of an immersion whose mean curvature 
is constant. The special case 
is a minimal surface.
In 3-dimensional Euclidean space, the plane,
sphere, circular cylinder,
and catenoid are standard examples, with the plane
and catenoid belonging to the minimal case.
For a conformal immersion of an oriented surface in a 3-dimensional manifold of constant sectional
curvature, the Hopf differential is holomorphic
precisely when 
is constant. This connects constant mean curvature surfaces
with Riemann surfaces, harmonic
maps, and integrable differential equations.
Hopf's theorem states that an immersed constant mean curvature sphere
in a complete, simply connected 3-dimensional
manifold of constant sectional
curvature is a round sphere. Alexandrov's theorem
states that every compact embedded surface of
constant mean curvature in 3-dimensional Euclidean
space is a round sphere.
