| In this thesis, we investigate hypersurface Mnr (r= 0,1) of pseudo-Euclidean space E1n+1+1 satisfying Δ→H= λH→, and show that if Mrn has at most three distinct principal curvatures, then it has constant mean curvature.In order to complete the proof of main theorem, according to the forms of the shape operator A, we discuss different cases. For r= 0, the shape operator A of Mn is always diagonalizable. But for r=1, the shape operator A of M1n has three nondiagonalizable forms:(Ⅱ), (Ⅲ) and (Ⅳ).In section 2, we study hypersurface Mrn with diagonalizable shape operator, and in this case, prove that if Mrn satisfying Δ→H=λH→ has at most three distinct principal curvatures, then it has constant mean curvature.In section 3, section 4 and section 5, we consider the three cases that the shape operator of M1n has the forms (Ⅱ), (Ⅲ) and (Ⅳ) separately, and get similar results. |
