Nonexistence of the NNSC-cobordism of Bartnik data

作     者：Bo Leyang Shi Yuguang Leyang Bo;Yuguang Shi

作者机构：Key Laboratory of Pure and Applied MathematicsSchool of Mathematical SciencesPeking UniversityBeijing 100871China 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2021年第64卷第7期

页      面：1357-1372页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：supported by National Natural Science Foundation of China(National Key R&D Program of China)(Grant No.11731001) Postdoctoral Science Foundation of China(Grant No.2020M680171) 

主　　题：cobordism scalar curvature mean curvature 

摘      要：In this paper,we consider the problem of the nonnegative scalar curvature(NNSC)-cobordism of Bartnik data(∑_(1)^(n-1),γ_(1),H_(1))and(∑_(2)^(n-1),γ_(2),H_(2)).We prove that given two metricsγ_(1)andγ_(2)on S^(n-1)(3≤n≤7)with H_(1)fixed,then(S^(n-1),γ_(1),H_(1))and(S^(n-1),γ_(2),H_(2))admit no NNSC-cobordism provided the prescribed mean curvature H2 is large enough(see Theorem 1.3).Moreover,we show that for n=3,a much weaker condition that the total mean curvature∫_(s^(2))H_(2)dpγ_(2)is large enough rules out NNSC-cobordisms(see Theorem 1.2);if we require the Gaussian curvature ofγ_(2)to be positive,we get a criterion for nonexistence of the trivial NNSCcobordism by using the Hawking mass and the Brown-York mass(see Theorem 1.1).For the general topology case,we prove that(∑_(1)^(n-1),γ_(1),0)and(∑_(2)^(n-1),γ_(2),H_(2))admit no NNSC-cobordism provided the prescribed mean curvature H_(2)is large enough(see Theorem 1.5).