The paper quotes the concept of Ricci curvature decay to zero. Base on this new concept, by modifying the proof of the canonical Cheeger-Gromoll Splitting Theorem, the paper proves that for a complete non-compact Riem...The paper quotes the concept of Ricci curvature decay to zero. Base on this new concept, by modifying the proof of the canonical Cheeger-Gromoll Splitting Theorem, the paper proves that for a complete non-compact Riemannian manifold M with Ricci curvature decay to zero, if there is a line in M, then the isometrically splitting M = R × N is true.展开更多
The paper studies a class of nonlinear elliptic partial differential equations on a compact Riemannian manifold （M,g） with some curvature restriction. The authors try to prove some uniqueness and nonexistent results...The paper studies a class of nonlinear elliptic partial differential equations on a compact Riemannian manifold （M,g） with some curvature restriction. The authors try to prove some uniqueness and nonexistent results for the positive solutions of the equations concerned.展开更多
In a previous paper(Jiang and Yang(2021)),we constructed complete manifolds of positive Ricci curvature with quadratically asymptotically nonnegative curvature and infinite topological type but dimensions greater than...In a previous paper(Jiang and Yang(2021)),we constructed complete manifolds of positive Ricci curvature with quadratically asymptotically nonnegative curvature and infinite topological type but dimensions greater than or equal to 6.The purpose of the present paper is to use a different technique to exhibit a family of complete I-dimensional(I≥5)Riemannian manifolds of positive Ricci curvature,quadratically asymptotically nonnegative sectional curvature,and certain infinite Betti numbers bj(2≤j≤I-2).展开更多
We give a survey on 4-dimensional manifolds with positive isotropic curvature. We will introduce the work of B. L. Chen, S. H. Tang and X. P. Zhu on a complete classification theorem on compact four-manifolds with pos...We give a survey on 4-dimensional manifolds with positive isotropic curvature. We will introduce the work of B. L. Chen, S. H. Tang and X. P. Zhu on a complete classification theorem on compact four-manifolds with positive isotropic curvature （PIC）. Then we review an application of the classification theorem, which is from Chen and Zhu＇s work. Finally, we discuss our recent result on the path-connectedness of the moduli spaces of Riemannian metrics with positive isotropic curvature.展开更多
The author gives an optimum estimate of the first eigenvalue of a compact Riemannian manifold. It is shown that let M be a compact Riemannian manifold, then the first eigenvalue λ<sub>1</sub> of the Lapla...The author gives an optimum estimate of the first eigenvalue of a compact Riemannian manifold. It is shown that let M be a compact Riemannian manifold, then the first eigenvalue λ<sub>1</sub> of the Laplace operator of M satisfies α<sub>1</sub>+max{0,-（n-1）K}≥π<sup>2</sup>/d<sup>2</sup> where d is the diameter of M and （n-1）K is the negative lower bound of the Ricci curvature of M.展开更多
In this paper, the authors consider the problem of which(generalized) momentangle manifolds admit Ricci positive metrics. For a simple polytope P, the authors can cut off one vertex v of P to get another simple polyto...In this paper, the authors consider the problem of which(generalized) momentangle manifolds admit Ricci positive metrics. For a simple polytope P, the authors can cut off one vertex v of P to get another simple polytope P_v, and prove that if the generalized moment-angle manifold corresponding to P admits a Ricci positive metric, the generalized moment-angle manifold corresponding to P_v also admits a Ricci positive metric. For a special class of polytope called Fano polytopes, the authors prove that the moment-angle manifolds corresponding to Fano polytopes admit Ricci positive metrics. Finally some conjectures on this problem are given.展开更多
文摘The paper quotes the concept of Ricci curvature decay to zero. Base on this new concept, by modifying the proof of the canonical Cheeger-Gromoll Splitting Theorem, the paper proves that for a complete non-compact Riemannian manifold M with Ricci curvature decay to zero, if there is a line in M, then the isometrically splitting M = R × N is true.
文摘The paper studies a class of nonlinear elliptic partial differential equations on a compact Riemannian manifold （M,g） with some curvature restriction. The authors try to prove some uniqueness and nonexistent results for the positive solutions of the equations concerned.
基金supported by National Natural Science Foundation of China(Grant Nos.11571228 and 12071283)fund of Shanghai Normal University(Grant No.SK202002)。
文摘In a previous paper(Jiang and Yang(2021)),we constructed complete manifolds of positive Ricci curvature with quadratically asymptotically nonnegative curvature and infinite topological type but dimensions greater than or equal to 6.The purpose of the present paper is to use a different technique to exhibit a family of complete I-dimensional(I≥5)Riemannian manifolds of positive Ricci curvature,quadratically asymptotically nonnegative sectional curvature,and certain infinite Betti numbers bj(2≤j≤I-2).
基金Acknowledgements The first author was partially supported by the National Natural Science Foundation of China （Grant Nos. 11025107, 11521101） and a grant （No. 141gzd02） from Sun Yat-sen University.
文摘We give a survey on 4-dimensional manifolds with positive isotropic curvature. We will introduce the work of B. L. Chen, S. H. Tang and X. P. Zhu on a complete classification theorem on compact four-manifolds with positive isotropic curvature （PIC）. Then we review an application of the classification theorem, which is from Chen and Zhu＇s work. Finally, we discuss our recent result on the path-connectedness of the moduli spaces of Riemannian metrics with positive isotropic curvature.
文摘The author gives an optimum estimate of the first eigenvalue of a compact Riemannian manifold. It is shown that let M be a compact Riemannian manifold, then the first eigenvalue λ<sub>1</sub> of the Laplace operator of M satisfies α<sub>1</sub>+max{0,-（n-1）K}≥π<sup>2</sup>/d<sup>2</sup> where d is the diameter of M and （n-1）K is the negative lower bound of the Ricci curvature of M.
基金supported by the National Natural Science Foundation of China(Nos.11471167,11571186,11701411,11801580)
文摘In this paper, the authors consider the problem of which(generalized) momentangle manifolds admit Ricci positive metrics. For a simple polytope P, the authors can cut off one vertex v of P to get another simple polytope P_v, and prove that if the generalized moment-angle manifold corresponding to P admits a Ricci positive metric, the generalized moment-angle manifold corresponding to P_v also admits a Ricci positive metric. For a special class of polytope called Fano polytopes, the authors prove that the moment-angle manifolds corresponding to Fano polytopes admit Ricci positive metrics. Finally some conjectures on this problem are given.
