第二基本型的一类抽象光滑泛函的变分问题

VARIATIONAL PROBLEM OF ONE ABSTRACT SMOOTH FUNCTIONAL ABOUT SECOND FUNDAMENTAL FORM

假设φ:M^(n)→N_(n+p)是一般外围流形中的n维子流形,S是该子流形的第二基本型模长的平方,本文构造了S的一-类幂函数型泛函G(_(n,F))(φ)=∫_(M)F(S)dv,其中F:[0,∞)→R为一光滑抽象函数.此泛函抽象刻画了子流形与全测地子流形的差异,并且与Willmore猜想有着密切联系.本文计算了该泛函的第一变分公式,并在单位球面中构造了该泛函临界点的一些例子,进一步,基于两个著名的矩阵不等式,我们推导了泛函临界点的Simons型积分不等式,并基于此给出了间隙现象的讨论.

For an n-dimensional submanifold in a general real ambient manifoldφ:M_(n)→N^(n+p),let S denote the square length of second fundamental form ofφ.In this paper,we introduce one abstract functional concerning S as G(_(n,F))(φ)=R M F(S)dv,where F:[0,∞)→R is a smooth abstract function,which measures abstractly how derivationsφ(M)from a totally geodesic submanifold and has a closed relation with the well-known Willmore conjecture.For this functional,the rst variational equation is obtained,and in unit sphere,we construct a few examples of critical points.Moreover,by two famous matrix inequalities,we derive out the Simons'type integral inequalities,and based on which some gap phenomenon have been classified.