Studies On Vanishing Theorems And Rigidity Theorems Of Submanifolds

In this paper,we mainly study the existence of nontrivial harmonic P-forms on com-plete non-compact submanifold Mn(n ? 3)immersed in a Hadamard manifold,and the rigidity theorems of the hypersurfaces in Euclidean space RN+1,hyperbolic space Hn+1 or the open hemisphere S+n+1.The main content consists of two parts:Chapter 2 and Chapter 3.In Chapter 2,we consider the vanishing theorems of harmonic p-forms on complete non-compact submanifold Mn(n ? 3)immersed in a Hadamard manifold.Firstly,when M has(Pp)property and finite total curvature,by applying an estimate for the Weitzen-bock curvature operator in Lin[27],Hoffman-Spruck inequality,Kato inequality and Morse iteration,we prove that the space of the L2 harmonic p-forms on M has finite dimensions,which is an extension of Theorem 3.3 in Lin[27].In addition,under the assumption that the submanifolds have suff1ciently small total curvature,we obtain that all L2 harmonic p-forms must be trivial,which extends the results about Euclidean space of Lin[29].Fi-nally,if M has finite total curvature and at most Euclidean volume growth,we investigate that all Lq harmonic p-forms must be trivial for any 2?p?n-2,2<q<4n/2n-1.In Chapter 3,the rigidity theorems of the hypersurfaces in Euclidean space Rn+1,hyperbolic space Hn+1 or the open hemisphere S+n+1 are discussed.Firstly,by applying Hsiung-MinKowski formula from Kwong[18],we prove that a oriented,closed and convex hypersurface ? immersed in the above semi-Riemannian manifolds must be a geodesic hypersphere,if there are some linear relations between higher order mean curvatures on?.Besides,we show an analogous result holds if ? is a closed,oriented and convex hypersurface embedded in the above semi-Riemannian manifolds.