Symmetry via spherical reflection

Abstract

In 1955 Alexandrov [2, 7, p. 147] introduced the method of planar reflection as a procedure for proving symmetry of closed surfaces of constant mean curvature. In subsequent years, a number of refinements were given, and the method was applied successfully to other problems in differential geometry and partial differential equations, cf. [10, 5, 12, 9, 3, 12]. In the present paper, we introduce a new procedure for obtaining symmetry which may be considered an extension of the planar reflection method to reflection about spheres.

In the interest of clearly delineating the new features of the method, in this initial presentation we give a new proof of the original Alexandrov theorem: A closed embedded surface of constant mean curvature in Euclidean space is a round sphere. Our proof of this result is not simpler than that of Alexandrov; it is, however, different in some crucial respects.

Our justification for introducing the new procedure lies in its applicability to configurations that are not accessible to planar reflection arguments. In particular, we use the new method in [8] to prove the non-existence in certain cases of embedded liquid bridges joining two planes that meet to form a wedge domain, a result whose interest derives in turn largely from the fact that under the same (physical) boundary conditions embedded bridges do exist and can be stable when the planes are parallel.

The method of spherical reflection as presented below relies on a new maximum principle related to surface geometry, which we believe to have independent interest. Further, the method utilizes a peculiar characterization of spherical surfaces by aggregate symmetry, which should also have interest for other problems arising in the global differential geometry of surfaces.