Conformal Geometry of Hypersurfaces in Lorentz Space Forms

Abstract EN

Let x:Mn→M1n+1(c) be a space-like hypersurface without umbilical points in the Lorentz space form M1n+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 M1n+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 H1n+1(-1).