Geometry Of Hypersurfaces And Minimal Lagrangian Surfaces

In this paper,we mainly studied geometry of hypersurfaces in complex projective space and the geometric properties of the minimal Lagrangian surfaces in the product space H²×H²of two hyperbolic planes,especially the properties related to variational stability.A hypersurface of constant mean curvature in a Riemannian manifold is a critical point of the area functional under any volume preserving variation.If,furthermore,the second variational of the area functional is non-negative,then the hypersurface is called a stable constant mean curvature hypersurface.We prove that the four types of homogeneous hypersurfaces B,C,D and E classified by Takagi in complex projective space are unstable as hypersurfaces with constant mean curvature.The weighted minimal hypersurface in a Riemann manifold is the critical point of the weighted area functional and it is one of generalizations of minimal hypersurfaces.A weighted minimal hypersurface with non-negative second variation of the weighted area functional is said to be stable.For the stable weighted minimal hypersurfaces in weighted Riemann manifolds,we obtain the diameter estimation.The Minkowski integral formula for hypersurfaces in real space forms has many important applications.In the complex space forms,by using the gradient vector field of the distance function,we obtain higher order Minkowski type integral formulas,which extend the first two Minkowski integral formulas given by V.Martuno and G.Tralli^[1].We also get some integral inequalities.Finally,we discuss the geometric properties of Lagrange surfaces in H²×H²and give the classification of minimal Lagrange surfaces under certain conditions.