The properties of the first eigenvalue of a class of (<em>p</em>,<em>q</em>) Laplacian are investigated. A variational formulation for the first eigenvalue of the Laplacian on a closed Riemanni...The properties of the first eigenvalue of a class of (<em>p</em>,<em>q</em>) Laplacian are investigated. A variational formulation for the first eigenvalue of the Laplacian on a closed Riemannian manifold is obtained. This eigenvalue corresponds to a nonlinear, coupled system of <em>p</em>-Laplacian partial differential equations. The main idea is to investigate the evolution of the first eigenvalue of the system under the Ricci harmonic flow. It is also possible to construct monotonic quantities based on them and study their evolution which is done.展开更多
We prove that for a compact Finsler manifold M with nonnegative weighted Ricci curvature,if its first closed(resp.Neumann)eigenvalue of Finsler-Laplacian attains the sharp lower bound,then M is isometric to a circle(r...We prove that for a compact Finsler manifold M with nonnegative weighted Ricci curvature,if its first closed(resp.Neumann)eigenvalue of Finsler-Laplacian attains the sharp lower bound,then M is isometric to a circle(resp.a segment).Moreover,a lower bound of the first eigenvalue of Finsler-Laplacian with Dirichlet boundary condition is also estimated.These generalize the corresponding results in recent literature.展开更多
This paper discusses the first eigenvalue on a compact Riemann manifold with the negative lower bound Ricci curvature. Let M be a compact Riemann manifold with the Ricci curvature≥-R, R=const. ≥0 and d is the diamet...This paper discusses the first eigenvalue on a compact Riemann manifold with the negative lower bound Ricci curvature. Let M be a compact Riemann manifold with the Ricci curvature≥-R, R=const. ≥0 and d is the diameter of M. Our main result is that the first eigenvalue λ1 of M satisfies λ1≥π^2/d^2-0.518R.展开更多
In this paper,we discuss the lower bound of the first eigenvalue of a closed convex or a minimal hypersurface in a closed Riemann manifold with Ricci curvature bounded below by a positive number.
In this letter, we deal with the following problem -△u= λu on n-dimensional compact Ricmann manifold M and obtain two different estimates of the first nonzero eigenvalue when Ricci curvature of M has a negative lowe...In this letter, we deal with the following problem -△u= λu on n-dimensional compact Ricmann manifold M and obtain two different estimates of the first nonzero eigenvalue when Ricci curvature of M has a negative lower bound. The results are improvements of the展开更多
We obtain the Laplacian comparison theorem and the Bishop- Gromov comparison theorem on a Finsler manifold with the weighted Ricci curvature Ric∞ bounded below. As applications, we prove that if the weighted Ricci cu...We obtain the Laplacian comparison theorem and the Bishop- Gromov comparison theorem on a Finsler manifold with the weighted Ricci curvature Ric∞ bounded below. As applications, we prove that if the weighted Ricci curvature Ri∞ is bounded below by a positive number, then the manifold must have finite fundamental group, and must be compact if the distortion is also bounded. Moreover, we give the Calabi-Yau linear volume growth theorem on a Finsler manifold with nonnegative weighted Ricci curvature.展开更多
This paper proves that the first eigenfunctions of the Finsler p-Lapalcian are C^(1,α). Using a gradient comparison theorem and one-dimensional model, we obtain the sharp lower bound of the first Neumann and closed e...This paper proves that the first eigenfunctions of the Finsler p-Lapalcian are C^(1,α). Using a gradient comparison theorem and one-dimensional model, we obtain the sharp lower bound of the first Neumann and closed eigenvalue of the p-Laplacian on a compact Finsler manifold with nonnegative weighted Ricci curvature,on which a lower bound of the first Dirichlet eigenvalue of the p-Laplacian is also obtained.展开更多
In this paper, we obtain an estimate for the lower bound for the dimensions of harmonic functions with polynomial growth and a Liouville type theorem on manifolds with nonnegative Ricci curvature whose tangent cone at...In this paper, we obtain an estimate for the lower bound for the dimensions of harmonic functions with polynomial growth and a Liouville type theorem on manifolds with nonnegative Ricci curvature whose tangent cone at infinity is a unique metric cone with a conic measure.展开更多
In this paper, we obtain Chen’s inequalities in (k,?μ)-contact space form with a semi-symmetric non-metric connection. Also we obtain the inequalites for Ricci and K-Ricci curvatures.
This paper aims at proving a conjecture posed by S. T. Yau: Let M be an m-dimen-sional compact Riemann manifold with the Ricci curvature≥-R, where R= const.≥0. Suppose d is the diameter of M and λ1 is the first eig...This paper aims at proving a conjecture posed by S. T. Yau: Let M be an m-dimen-sional compact Riemann manifold with the Ricci curvature≥-R, where R= const.≥0. Suppose d is the diameter of M and λ1 is the first eigenvalue of M. Then there exists a constant Cm dependent only on m such展开更多
文摘The properties of the first eigenvalue of a class of (<em>p</em>,<em>q</em>) Laplacian are investigated. A variational formulation for the first eigenvalue of the Laplacian on a closed Riemannian manifold is obtained. This eigenvalue corresponds to a nonlinear, coupled system of <em>p</em>-Laplacian partial differential equations. The main idea is to investigate the evolution of the first eigenvalue of the system under the Ricci harmonic flow. It is also possible to construct monotonic quantities based on them and study their evolution which is done.
基金supported by National Natural Science Foundation of China(Grant No.11171253)the Natural Science Foundation of Ministry of Education of Anhui Province(Grant No.KJ2012B197)
文摘We prove that for a compact Finsler manifold M with nonnegative weighted Ricci curvature,if its first closed(resp.Neumann)eigenvalue of Finsler-Laplacian attains the sharp lower bound,then M is isometric to a circle(resp.a segment).Moreover,a lower bound of the first eigenvalue of Finsler-Laplacian with Dirichlet boundary condition is also estimated.These generalize the corresponding results in recent literature.
基金Supported by the NNSF of China(10271011)Supported by LIMB of the Ministry of Education China
文摘This paper discusses the first eigenvalue on a compact Riemann manifold with the negative lower bound Ricci curvature. Let M be a compact Riemann manifold with the Ricci curvature≥-R, R=const. ≥0 and d is the diameter of M. Our main result is that the first eigenvalue λ1 of M satisfies λ1≥π^2/d^2-0.518R.
文摘In this paper,we discuss the lower bound of the first eigenvalue of a closed convex or a minimal hypersurface in a closed Riemann manifold with Ricci curvature bounded below by a positive number.
文摘In this letter, we deal with the following problem -△u= λu on n-dimensional compact Ricmann manifold M and obtain two different estimates of the first nonzero eigenvalue when Ricci curvature of M has a negative lower bound. The results are improvements of the
基金This work was supported in part by the Natural Science Foundation of Anhui Province （No. 1608085MA03） and the National Natural Science Foundation of China （Grant No. 11471246）.
文摘We obtain the Laplacian comparison theorem and the Bishop- Gromov comparison theorem on a Finsler manifold with the weighted Ricci curvature Ric∞ bounded below. As applications, we prove that if the weighted Ricci curvature Ri∞ is bounded below by a positive number, then the manifold must have finite fundamental group, and must be compact if the distortion is also bounded. Moreover, we give the Calabi-Yau linear volume growth theorem on a Finsler manifold with nonnegative weighted Ricci curvature.
基金Supported by the National Natural Science Foundation of China(11371386)the European Union’s Seventh Framework Programme(FP7/2007-2013)under grant agreement(317721)
基金supported by National Natural Science Foundation of China (Grant No. 11471246)Natural Science Foundation of Higher Education in Anhui Province (Grant No. KJ2014A257)
文摘This paper proves that the first eigenfunctions of the Finsler p-Lapalcian are C^(1,α). Using a gradient comparison theorem and one-dimensional model, we obtain the sharp lower bound of the first Neumann and closed eigenvalue of the p-Laplacian on a compact Finsler manifold with nonnegative weighted Ricci curvature,on which a lower bound of the first Dirichlet eigenvalue of the p-Laplacian is also obtained.
基金partially supported by NSFC(11701580and 11521101)the Fundamental Research Funds for the Central Universities(17lgpy13)
文摘In this paper, we obtain an estimate for the lower bound for the dimensions of harmonic functions with polynomial growth and a Liouville type theorem on manifolds with nonnegative Ricci curvature whose tangent cone at infinity is a unique metric cone with a conic measure.
基金Supported by the Study of Some Problems on Holomorphic Function Space and Operator Theory(11671357)the National Natural Science Foundation of China(11771139)+1 种基金the Natural Science Foundation of Guangxi(2015jjBA10049)partially supported by the Hu Guozan Study-Abroad Grant for graduates(China)for her visit to UC Irvine in 2015–2016 when part of this work was done
文摘We give the sharp lower bound for Ricci curvature on the real ellipsoid in Cn＋l,and prove the Lichnerowicz-Obata theorem for Kohn Laplacian.
文摘In this paper, we obtain Chen’s inequalities in (k,?μ)-contact space form with a semi-symmetric non-metric connection. Also we obtain the inequalites for Ricci and K-Ricci curvatures.
基金Partially supported by the National Natural Science Foundation of China
文摘This paper aims at proving a conjecture posed by S. T. Yau: Let M be an m-dimen-sional compact Riemann manifold with the Ricci curvature≥-R, where R= const.≥0. Suppose d is the diameter of M and λ1 is the first eigenvalue of M. Then there exists a constant Cm dependent only on m such
