Conformal Geometry of Hypersurfaces in Lorentz Space Forms. (16th September 2013) - Li, Tongzhu; Nie, Changxiong
Abstract : Let x : M n → M 1 n + 1 ( c ) be a space-like hypersurface without umbilical points in the Lorentz space form M 1 n + 1 ( c ) . We define the conformal metric and the conformal second fundamental form on the hypersurface, which determines the hypersurface up to conformal transformation of M 1 n + 1 ( c ) . We calculate the Euler-Lagrange equations of the volume functional of the hypersurface with respect to the conformal metric, whose critical point is called a Willmore hypersurface, and we give a conformal characteristic of the hypersurfaces with constant mean curvature and constant scalar curvature. Finally, we prove that if the hypersurface x with constant mean curvature and constant scalar curvature is Willmore, then x is a hypersurface in H 1 n + 1 ( - 1 ) .