Conformal Geometry of Hypersurfaces in Lorentz Space Forms

Let be a space-like hypersurface without umbilical points in the Lorentz space form . We define the conformal metric and the conformal second fundamental form on the hypersurface, which determines the hypersurface up to conformal transformation of . 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 with constant mean curvature and constant scalar curvature is Willmore, then is a hypersurface in . 1. Introduction Let be an immersed submanifold in sphere . In [1], on the submanifold the Wang has constructed a complete invariant system of the M？bius transformation group of . Especially for the hypersurface, the M？bius invariants, the M？bius metric, and the M？bius second fundamental form determine the hypersurface up to M？bius transformations provided the dimension of hypersurface (also see [2]). After that, the study of the M？bius geometry has been a topic of increasing interest (see [3–6]). In this paper we study space-like hypersurfaces in the Lorentz space form under the conformal transformation group. We follow Wang’s idea and construct conformal invariants of space-like hypersurfaces which determine hypersurfaces up to a conformal transformation. For the Lorentz space form, there exists a united conformal compactification , which is the projectivized light cone in induced from (see [7, 8]). Using conformal compactification , we define the conformal metric and the conformal second fundamental form on a hypersurface in the Lorentz space form, which determines a hypersurface up to a conformal transformation. Clearly, the volume functional with respect to the conformal metric is a conformal invariant. We call a critical hypersurface of the volume functional Willmore hypersurface. There are many studies about the Willmore hypersurface in the Lorentz space form (see, [7, 9, 10]). Our main goal is to calculate the Euler-Lagrange equation for the volume functional by conformal invariants and to find some special Willmore hypersurfaces. We find that maximal hypersurfaces in Lorentz space form are not Willmore in general if the dimension . We give a conformal characteristic of the hypersurfaces with constant mean curvature and constant scalar curvature. By the conformal characteristic, we prove that if the hypersurfaces are Willmore, then the hypersurfaces must be in .