Alexandrov reflection, also called the method of moving planes, is a method of proving symmetry by comparing an object with its reflection across a moving plane. In differential geometry, it uses a comparison principle for surfaces with prescribed mean curvature (Alexandrov 1962, Meeks et al. 2016). The method also proves symmetry of solutions of elliptic partial differential equations by comparing a solution with its reflected values (Gidas et al. 1979).
For a connected, compact, smooth embedded surface without boundary and with constant mean curvature in three-dimensional Euclidean space, begin with a plane outside the enclosed region. Move it through a family of parallel planes, reflecting the cap cut off by the moving plane. Continue while the reflected cap lies inside the enclosed region. At the limiting position, the interior maximum principle applies to contact away from the moving plane, and the boundary maximum principle handles contact on it. These principles require matching normal vectors, the same signed mean curvature, and a one-sided comparison. They give local coincidence, which extends by connectedness to a plane of symmetry. Repeating the argument in every direction proves Alexandrov's soap bubble theorem (Meeks et al. 2016).
