Plateau's rotating drops and rotational figures of equilibrium

We give a detailed classification of all rotationally symmetric figures of equilibrium corresponding to rotating liquid masses subject to surface tension. When the rotation rate is zero, these shapes were studied by Delaunay who found six different qualitative types of complete connected interfaces (spheres, cylinders, unduloids, nodoids, catenoids, and planes). We find twenty-six qualitatively different interfaces providing a complete picture of symmetric equilibrium shapes, some of which have been studied by other authors. In particular, combining our work with that of Beer, Chandrasekhar, Gulliver, Smith, and Ross, we conclude that every compact equilibrium is in either a smooth connected one parameter family of spheroids or a smooth connected one parameter family of tori (possibly immersed in either case).