Submanifold Of Riemannian Manifold And A Parabolic Equation On Riemannian Manifold

In the paper, we mainly investigate rigidity theorem of the oriented compact minimal subman-ifold in local symmetry Riemannian manifold, we also study complete hypersurfaces with constantmean curvature and Lδfinite index in hyperbolic spaces, and study gradient estimate of a parabolicequation on Riemannian manifold. The paper consists of four chapters.In chapter1, we introduce some related definitions, results and prerequisite knowledge thatwill be used in this paper.In chapter2, we study rigidity theorem of the oriented compact minimal submanifold in localsymmetry Riemannian manifold, by using a matrix inequality, we got a rigidity theorem of thiskind of submanifold, the result that we get partly improve a conclusion.In chapter3, we study complete hypersurface with constant mean curvature in Riemannianmanifold. Under certain conditions, we get this kind of submanifold must be compact.In chapter4, we study a parabolic equation on Riemannian manifold, and get a gradientestimate of positive solution, and using the estimate we get a Liouville theorem.