It is proved that if M^n is an n-dimensional complete submanifold with parallel mean curvature vector and flat normal bundle in S^n＋p（1）, and if supM S 〈 α（n, H）, where α（n,H）=n＋n^3/2（n-1）H^2-n（n-2）/n（...It is proved that if M^n is an n-dimensional complete submanifold with parallel mean curvature vector and flat normal bundle in S^n＋p（1）, and if supM S 〈 α（n, H）, where α（n,H）=n＋n^3/2（n-1）H^2-n（n-2）/n（n-1）√n^2H^4＋4（n-1）H^2,then M^n must be the totally urnbilical sphere S^n（1/√1＋H^2）.An example to show that the pinching constant α（n, H） appears optimal is given.展开更多
Let K ( ) Rn be a convex body of volume 1 whose barycenter is at the origin,LK be the isotropic constant of K. Finding the least upper bound of LK, being called Bourgain's problem, is a well known open problem in ...Let K ( ) Rn be a convex body of volume 1 whose barycenter is at the origin,LK be the isotropic constant of K. Finding the least upper bound of LK, being called Bourgain's problem, is a well known open problem in the local theory of Banach space.The best estimate known today is LK ＜ cn1/4 log n, recently shown by Bourgain, for an arbitrary convex body in any finite dimension. Utilizing the method of spherical section function, it is proven that if K is a convex body with volume 1 and r1Bn2 ( )K ( ) r2 Bn2, (r1 ≥1/2, r2 ≤ -√n/2), then1/√2πe ≤ LK ≤ 1/2√3,and find the conditions with equality. Further,the geometric characteristic of isotropic bodies is shown.展开更多
基金Research supported by the National Natural Science Foundation of China(10771187)Trans-Century Training Programme Foundation for Talents by the Ministry of Education of China.
文摘It is proved that if M^n is an n-dimensional complete submanifold with parallel mean curvature vector and flat normal bundle in S^n＋p（1）, and if supM S 〈 α（n, H）, where α（n,H）=n＋n^3/2（n-1）H^2-n（n-2）/n（n-1）√n^2H^4＋4（n-1）H^2,then M^n must be the totally urnbilical sphere S^n（1/√1＋H^2）.An example to show that the pinching constant α（n, H） appears optimal is given.
基金This work was supported by the National Natural Science Foundation of China(Grant No.10271071).
文摘Let K ( ) Rn be a convex body of volume 1 whose barycenter is at the origin,LK be the isotropic constant of K. Finding the least upper bound of LK, being called Bourgain's problem, is a well known open problem in the local theory of Banach space.The best estimate known today is LK ＜ cn1/4 log n, recently shown by Bourgain, for an arbitrary convex body in any finite dimension. Utilizing the method of spherical section function, it is proven that if K is a convex body with volume 1 and r1Bn2 ( )K ( ) r2 Bn2, (r1 ≥1/2, r2 ≤ -√n/2), then1/√2πe ≤ LK ≤ 1/2√3,and find the conditions with equality. Further,the geometric characteristic of isotropic bodies is shown.
