Hypersurfaces in Warped Products

Abstract

A classical result of Alexandrov [10] states that a compact hypersurface with constant mean curvature embedded in Euclidean space must be a round sphere. The original proof is based on a clever use of the maximum principle for elliptic partial differential equations. This method, now called the Alexandrov’s reflexion method, also works for hypersurfaces in ambient spaces having a sufficiently large number of isometric reflexions, for instance in the hyperbolic space. It is worth to observe that, in an analytical context, this is the root of what has been called the “moving plane” technique, initiated by the pioneering work of Serrin, [254], and over and over successfully applied to prove special symmetries of solutions to certain PDE’s.