In this paper, we prove that several different definitions of the Finsler-Laplacian are equivalent. Then we prove that any Berwald metric is affinely equivalent to its mean metric and give some upper or lower bound es...In this paper, we prove that several different definitions of the Finsler-Laplacian are equivalent. Then we prove that any Berwald metric is affinely equivalent to its mean metric and give some upper or lower bound estimates for the first eigenvalue of the mean Laplacian on Berwald manifolds, which generalize some results in Riemannian geometry.展开更多
In this paper, the Laplacian on the holomorphic tangent bundle T 1,0 M of a complex manifold M endowed with a strongly pseudoconvex complex Finsler metric is defined and its explicit expression is obtained by using th...In this paper, the Laplacian on the holomorphic tangent bundle T 1,0 M of a complex manifold M endowed with a strongly pseudoconvex complex Finsler metric is defined and its explicit expression is obtained by using the Chern Finsler connection associated with (M, F ). Utilizing the initiated "Bochner technique", a vanishing theorem for vector fields on the holomorphic tangent bundle T 1,0 M is obtained.展开更多
In this paper, we give a sharp lower bound for the first(nonzero) p-eigenvalue on a compact Finsler manifold M without boundary or with convex boundary if the weighted Ricci curvature RicciNis bounded from below by a ...In this paper, we give a sharp lower bound for the first(nonzero) p-eigenvalue on a compact Finsler manifold M without boundary or with convex boundary if the weighted Ricci curvature RicciNis bounded from below by a constant K in terms of the diameter d of a manifold, dimension, K, p and N. In particular, if RicciNπis non-negative, then the first p-eigenvalue is bounded from below by(p-1)(~πp/d)p, and the equality holds if and only if M is either a circle or a segment.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
Cheng-type inequality, Cheeger-type inequality and Faber-Krahn-type inequality are generalized to Finsler manifolds. For a compact Finsler manifold with the weighted Ricci curvature bounded from below by a negative co...Cheng-type inequality, Cheeger-type inequality and Faber-Krahn-type inequality are generalized to Finsler manifolds. For a compact Finsler manifold with the weighted Ricci curvature bounded from below by a negative constant, Li-Yau's estimation of the first eigenvalue is also given.展开更多
A horizontal Hodge Laplacian operator □h is defined for Hermitian holomorphic vector bundles over PTM on Khler Finsler manifold,and the expression of □h is obtained explicitly in terms of horizontal covariant deri...A horizontal Hodge Laplacian operator □h is defined for Hermitian holomorphic vector bundles over PTM on Khler Finsler manifold,and the expression of □h is obtained explicitly in terms of horizontal covariant derivatives of the Chern-Finsler connection.The vanishing theorem is obtained by using the α_Hα_H-method on Kahler Finsler manifolds.展开更多
Funk metrics are a kind of important Finsler metrics with constant negative flag curvature. In this paper, it is proved that any isoparametric hypersurface in Funk spaces has at most two distinct principal curvatures....Funk metrics are a kind of important Finsler metrics with constant negative flag curvature. In this paper, it is proved that any isoparametric hypersurface in Funk spaces has at most two distinct principal curvatures. Moreover, a complete classification of isoparametric families in a Funk space is given.展开更多
文摘In this paper, we prove that several different definitions of the Finsler-Laplacian are equivalent. Then we prove that any Berwald metric is affinely equivalent to its mean metric and give some upper or lower bound estimates for the first eigenvalue of the mean Laplacian on Berwald manifolds, which generalize some results in Riemannian geometry.
基金Project Supported by the National Natural Science Foundation of China (Nos. 10871145, 10771174)the Doctoral Program Foundation of the Ministry of Education of China (No. 2009007Q110053)
文摘In this paper, the Laplacian on the holomorphic tangent bundle T 1,0 M of a complex manifold M endowed with a strongly pseudoconvex complex Finsler metric is defined and its explicit expression is obtained by using the Chern Finsler connection associated with (M, F ). Utilizing the initiated "Bochner technique", a vanishing theorem for vector fields on the holomorphic tangent bundle T 1,0 M is obtained.
基金supported by National Natural Science Foundation of China (Grant No. 11671352)Zhejiang Provincial National Science Foundation of China (Grant No. LY19A010021)
文摘In this paper, we give a sharp lower bound for the first(nonzero) p-eigenvalue on a compact Finsler manifold M without boundary or with convex boundary if the weighted Ricci curvature RicciNis bounded from below by a constant K in terms of the diameter d of a manifold, dimension, K, p and N. In particular, if RicciNπis non-negative, then the first p-eigenvalue is bounded from below by(p-1)(~πp/d)p, and the equality holds if and only if M is either a circle or a segment.
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 National Natural Science Foundation of China(Nos.11471246,11171253)the Natural Science Foundation of the Anhui Higher Education Institutions(No.KJ2014A257)
文摘Cheng-type inequality, Cheeger-type inequality and Faber-Krahn-type inequality are generalized to Finsler manifolds. For a compact Finsler manifold with the weighted Ricci curvature bounded from below by a negative constant, Li-Yau's estimation of the first eigenvalue is also given.
基金supported by the National Natural Science Foundation of China(No.11712777)the Scientific Research Foundation of Shanghai University of Engineering Science(No.E1-0501-14-0112)
文摘A horizontal Hodge Laplacian operator □h is defined for Hermitian holomorphic vector bundles over PTM on Khler Finsler manifold,and the expression of □h is obtained explicitly in terms of horizontal covariant derivatives of the Chern-Finsler connection.The vanishing theorem is obtained by using the α_Hα_H-method on Kahler Finsler manifolds.
基金supported by National Natural Science Foundation of China (Grant No. 11471246)Anhui Provincial Natural Science Foundation (Grant No. 1608085MA03)Natural Science Foundation of Higher Education in Anhui Province (Grant No. KJ2014A257)
文摘Funk metrics are a kind of important Finsler metrics with constant negative flag curvature. In this paper, it is proved that any isoparametric hypersurface in Funk spaces has at most two distinct principal curvatures. Moreover, a complete classification of isoparametric families in a Funk space is given.