子流形低阶曲率泛函的变分计算与间隙现象  

Variational Calculation and Gap Phenomena of Low Order Curvature Functional of Submanifolds

作　　者：刘进[1] Liu Jin(College of Systems Engineering,National University of Defense Technology,Changsha 410073,China)

机构地区：[1]国防科技大学系统工程学院,长沙410073

出　　处：《数学理论与应用》2023年第3期23-60,共38页Mathematical Theory and Applications

基　　金：湖南省自然科学基金(No.2021JJ30771);国家自然科学基金(No.11701565)资助。

摘　　要：设φ:M^(n)→N^(n+p)是一般外围流形中的n维紧致无边子流形.φ的第二基本型模长平方S、平均曲率模长平方H^(2)和迹零第二基本型模长平方ρ=S-nH^(2)等重要的低阶曲率分别刻画了全测地、极小、全脐等重要的几何性质.本文构造低阶曲率泛函L(I,n,F)(φ)=∫_(M F)(S,H^(2))dv,L(II,n,F)(φ)=∫_(M) F(ρ,H^(2))dv,其中F:[0,+∞)×[0,+∞)→R是一个抽象的充分光滑的双变量函数.这类泛函可刻画子流形与全测地子流形、极小子流形和全脐子流形的整体差异,将多类子流形泛函囊括在统一的框架之下,且与子流形中多类著名问题,如Willmore猜想,有着密切联系.本文将计算第一变分公式,在空间形式中构造临界点的一些例子,推导泛函临界点的积分不等式,并基于此对间隙现象进行讨论.Letφ:M^(n)→N^(n+p)be an ndimensional compact without boundary submanifold in a general real ambient manifold.Its three important low order curvatures:the square length S of second fundamental form,the square lengthH^(2)of mean curvature,and the square lengthρ=S−nH^(2)of trace zero second fundamental form,respectively describe the geometric properties of totally geodesic,minimal,and totally umbilical.Let F:[0,+∞)×[0,+∞)→R be an abstract smooth bivariate function.In this paper,we construct two functionals L_(I,n,F)(φ)=∫_(M)F(S,H^(2))dv and L(II,n,F)(φ)=∫_(M)F(ρ,H^(2))dv,which include some wellknown functionals as special cases,measure how derivationsφfrom totally geodesic,minimal,or totally umbilical submanifolds 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.

关 键 词：第二基本型 低阶曲率 间隙现象 积分不等式 临界点 

分 类 号：O186.1[理学—数学]