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.
作者
刘进
Liu Jin(College of Systems Engineering,National University of Defense Technology,Changsha 410073,China)
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。
关键词
第二基本型
低阶曲率
间隙现象
积分不等式
临界点
Second fundamental form
Low order curvature
Gap phenomenon
Integral inequality
Critical point
