Willmore spacelike submanifolds in an indefinite space form N q n + p ( c )

摘要

Let N q n + p ( c ) be an ( n + p ) -dimensional connected indefinite space form of index q ( 1 ≤ q ≤ p ) and of constant curvature c . Denote by φ : M → N q n + p ( c ) the n -dimensional spacelike submanifold in N q n + p ( c ) , φ : M → N q n + p ( c ) is called a Willmore spacelike submanifold in N q n + p ( c ) if it is a critical submanifold to the Willmore functional W ( φ ) = ∫ M ρ n d v = ∫ M ( S - n H 2 ) n 2 d v , where S and H denote the norm square of the second fundamental form and the mean curvature of M and ρ 2 = S - n H 2 . If q = p , in , we proved some integral inequalities of Simons’ type and rigidity theorems for n -dimensional Willmore spacelike submanifolds in a Lorentzian space form N p n + p ( c ) . In this paper, we continue to study this topic and prove some integral inequalities of Simons’ type and rigidity theorems for n -dimensional Willmore spacelike submanifolds in an indefinite space form N q n + p ( c ) ( 1 ≤ q < p ).