This is the intersection of a coplanar constant mean curvature surface with its plane of symmetry; see papers by N. Korevaar, R. Kusner, and myself.
My research lies at the intersection of geometry and analysis. For instance, you might ask how much land you can enclose with 100 miles of fencing; this is the classical isoperimetric problem, which people have studied for over 2000 years. If you have a very large, flat field, the optimal shape is the first one you'd guess: a circle. However, the answer changes if, say, the land is a narrow strip between two forests, or if there's a river running through the field, or if the land isn't flat. (Is it better to surround the top of a hill or the pass between two peaks?) The best shape will satisfy a differential equation, and the solutions to this equation depend very much on the geometry of the land. Problems like that of finding isoperimetric regions form a rich area of study, with connections to other areas of mathematics such as function theory, probability, and topology, and even some applications.