关键词: 第二基本型, 低阶曲率, 间隙现象 , 积分不等式, 临界点

Abstract: Let $\varphi:M^{n}\to N^{n+p}$ be an $n$-dimensional compact without boundary sub-manifold in a general real ambient manifold. Its three important low order curvatures: the square length $S$ of second fundamental form, the square length $H^{2}$ of mean curvature, and the square length $\rho=S-nH^{2}$ of trace zero second fundamental form, respectively describe the geometric properties of totally geodesic, minimal, and totally umbilical. Let $F: [0,+\infty)\times [0,+\infty)\to \mathbb{R}$ be an abstract smooth bivariate function. In this paper, we construct two functionals ${\mathcal L}_{(I,n,F)}(\varphi)=\int_{M}F(S,H^{2}){\rm{d}}v$ and $ {\mathcal L}_{(II,n,F)}(\varphi)=\int_{M}F(\rho,H^{2}){\rm{d}}v$, which include some well-known functionals as special cases, measure how derivations $\varphi$ from totally geodesic, minimal, or totally umbilical sub-manifolds globally, and have a closed relation to the Willmore conjecture. For these functionals, we obtain the first variational equations, and construct a few examples of critical points in space forms. Moreover, we derive out some integral inequalities, and based on which classify the gap phenomenon.

Key words: Second fundamental form, Low order curvature,   Gap phenomenon,   Integral inequality,   Critical point