Constant Mean Curvature Hypersurfaces In Riemannian Manifolds

In this thesis, we study the constant mean curvature hypersurfaces in Riemannian man-ifolds. This paper consists of two parts.In the first part, we consider constant geodesic curvature curves on an oriented2-dimensional Riemannian manifold (M, g). We proof that if a sequence of constant geodesic curvature closed curves Γi,i∈N, with constant geodesic curvature kiâ†'+∞, converges to the point p∈M in the sense of Hausdorff distance, then the point p is a critical point of the Gauss curvature. Furthermore, given an initial point p and an initial speed v∈TpM, we investigate the behavior of the immersed(non-embedded) constant mean curvature curve γ(p,v,k) passing through p with speed v and large constant mean curvature k(?)0. We prove that, as k tends to∞, γ(p, v, k) converges in Hausdorff distance to the level curve of the Gauss curvature function passing through p. Still we study the existence of closed curve of degree d immersed in M with large constant curvature k.In the second part, we construct constant mean curvature hypersurfaces by gluing method. Assume n≥2, given a closed (n+1)-dimensional Riemannian manifold (M, g) and an oriented nondegenerate minimal hypersurface∑(?)M, we consider the connected sum of two copies of E in M at finitely many points satisfying some suitable assumptions. It is proved that, by this way we can construct constant mean curvature hypersurfaces with small constant mean curvature. More precisely, we show that the minimal hypersurface E is the multiplicity2limit of a family of constant mean curvature hypersurfaces whose topology degenerate as their mean curvature tends to0.