We prove that the index is bounded from below by a linear function of its first Betti number for any compact free boundary f-minimal hypersurface in certain positively curved weighted manifolds.
In order to study the Yamabe changing-sign problem,Bahri and Xu proposed a conjecture which is a universal inequality for p points in R^(m).They have verified the conjecture for p≤3.In this paper,we first simplify th...In order to study the Yamabe changing-sign problem,Bahri and Xu proposed a conjecture which is a universal inequality for p points in R^(m).They have verified the conjecture for p≤3.In this paper,we first simplify this conjecture by giving two sufficient and necessary conditions inductively.Then we prove the conjecture for the basic case m=1 with arbitrary p.In addition,for the cases when p=4,5 and m≥2,we manage to reduce them to the basic case m=1 and thus prove them as well.展开更多
Motivated by the theory of isoparametric hypersurfaces,we study submanifolds whose tubular hypersurfaces have some constant higher order mean curvatures.Here a k-th order mean curvature Q_k^v（k ≥ 1） of a submanifol...Motivated by the theory of isoparametric hypersurfaces,we study submanifolds whose tubular hypersurfaces have some constant higher order mean curvatures.Here a k-th order mean curvature Q_k^v（k ≥ 1） of a submanifold M^n-is defined as the k-th power sum of the principal curvatures,or equivalently,of the shape operator with respect to the unit normal vector v.We show that if all nearby tubular hypersurfaces of M have some constant higher order mean curvatures,then the submanifold M itself has some constant higher order mean curvatures Q_k^v independent of the choice of v.Many identities involving higher order mean curvatures and Jacobi operators on such submanifolds are also obtained.In particular,we generalize several classical results in isoparametric theory given by E.Cartan,K.Nomizu,H.F.Miinzner,Q.M.Wang,et al.As an application,we finally get a geometrical filtration for the focal submanifolds of isoparametric functions on a complete Riemannian manifold.展开更多
基金Supported by Beijing Natural Science Foundation(Grant No.Z190003)NSFC(Grant No.12171037)the Fundamental Research Funds for the Central Universities。
文摘We prove that the index is bounded from below by a linear function of its first Betti number for any compact free boundary f-minimal hypersurface in certain positively curved weighted manifolds.
基金partially supported by Beijing Natural Science Foundation(Grant No.Z190003)partially supported by National Sciences Foundation of USA(Grant No.DMS-19-08513)。
文摘In order to study the Yamabe changing-sign problem,Bahri and Xu proposed a conjecture which is a universal inequality for p points in R^(m).They have verified the conjecture for p≤3.In this paper,we first simplify this conjecture by giving two sufficient and necessary conditions inductively.Then we prove the conjecture for the basic case m=1 with arbitrary p.In addition,for the cases when p=4,5 and m≥2,we manage to reduce them to the basic case m=1 and thus prove them as well.
基金partially supported by NSFC(Grant No.11331002)the Fundamental Research Funds for the Central Universities
文摘Motivated by the theory of isoparametric hypersurfaces,we study submanifolds whose tubular hypersurfaces have some constant higher order mean curvatures.Here a k-th order mean curvature Q_k^v（k ≥ 1） of a submanifold M^n-is defined as the k-th power sum of the principal curvatures,or equivalently,of the shape operator with respect to the unit normal vector v.We show that if all nearby tubular hypersurfaces of M have some constant higher order mean curvatures,then the submanifold M itself has some constant higher order mean curvatures Q_k^v independent of the choice of v.Many identities involving higher order mean curvatures and Jacobi operators on such submanifolds are also obtained.In particular,we generalize several classical results in isoparametric theory given by E.Cartan,K.Nomizu,H.F.Miinzner,Q.M.Wang,et al.As an application,we finally get a geometrical filtration for the focal submanifolds of isoparametric functions on a complete Riemannian manifold.
