Conformal Geometry of Hypersurfaces in Lorentz Space Forms

Tongzhu Li,Changxiong Nie

doi:10.1155/2013/549602

Abstract

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 .

Highlights

Let x : Mn → Sn+p be an immersed submanifold in sphere Sn+k
For the hypersurface, the Mobius invariants, the Mobius metric, and the Mobius second fundamental form determine the hypersurface up to Mobius transformations provided the dimension of hypersurface n ≥ 3
The study of the Mobius geometry has been a topic of increasing interest

Summary

Introduction

Let x : Mn → Sn+p be an immersed submanifold in sphere Sn+k. In [1], on the submanifold the Wang has constructed a complete invariant system of the Mobius transformation group of Sn+p. In this paper we study space-like hypersurfaces in the Lorentz space form M1n+1(c) 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. Using conformal compactification Qn1+1, we define the conformal metric g and the conformal second fundamental form on a hypersurface in the Lorentz space form, which determines a hypersurface up to a conformal transformation. Our main goal is to calculate the Euler-Lagrange equation for the volume functional by conformal invariants and to find some special Willmore hypersurfaces.

Conformal Invariants of Hypersurfaces in Lorentz Space

The First Variation of the Conformal Volume Functional

Some Special Willmore Space-Like Hypersurfaces