Studies On Vanishing Theorems And Rigidity Theorems Of Submanifold

In this paper, we study the rigidity of linear Weingarten hypersurfaces in Lorentz space forms and hyperbolic space, and the vanishing theorems of harmonic 1-forms in submanifolds with property (Pp). The main content consists of two parts:Chapter 3 and Chapter 4.In Chapter 3, we consider the rigidity of hypersurfaces in Lorentz space forms and hyperbolic space. Firstly, a divergence lemma is proven for linear Weingarten spacelike hypersurfaces in Lorentz space forms, and by applying this lemma, we prove that hy-persurfaces must be totally umbilical under suitable restrictions on the Gauss map of hypersurfaces. This result generalises that of Aquino, Bezerra, Lima [11] and Aquino, Lima, Velasquez [15]. Similar results are obtained in Lorentz space forms for spacelike hypersurfaces, on which there are some linear relations between higher order mean curva-tures. Besides, we show analogous results hold if the ambient space is hyperbolic space.In Chapter 4, the vanishing theorems of nontrivial harmonic 1-forms in submanifolds with property (Pp) are discussed. Firstly, by applying Bochner-Weitzenbock formula, Kato inequality, Sobolev inequality and the estimation on Ricci curvature, under the assumption that the submanifolds are stable or have sufficiently small total curvature, we prove that such submanifolds have no nontrivial L2 harmonic 1-forms, which are extensions of the results of Kim and Yun [44] and Cavalcante, Mirandola, Vitorio [26]. In addition, if complete noncompact submanifolds are δ-stable or have sufficiently small total curvature, we obtain that all LP harmonic 1-forms must be trivial.