Displacement interpolations from a Hamiltonian point of view

One of the most well-known results in the theory of optimal transportation is the equivalence between the convexity of the entropy functional with respect to the Riemannian Wasserstein metric and the Ricci curvature lower bound of the underlying Riemannian manifold. There are also generalizations of this result to the Finsler manifolds and manifolds with a Ricci flow background. In this paper, we study displacement interpolations from the point of view of Hamiltonian systems and give a unifying approach to the above mentioned results.     

Uniform convexity and smoothness,and their applications in Finsler geometry

We generalize the Alexandrov–Toponogov comparison theorems to Finsler manifolds. Under suitable upper (lower, resp.) bounds on the flag and tangent curvatures together with the 2-uniform convexity (smoothness, resp.) of tangent spaces, we show the 2-uniform convexity (smoothness, resp.) of Finsler manifolds. As applications, we prove the almost everywhere existence of the second order differentials of semi-convex functions and of c-concave functions with the quadratic cost function.     

Comparison theorems in Finsler geometry and their applications

We prove Hessian comparison theorems, Laplacian comparison theorems and volume comparison theorems for Finsler manifolds under various curvature conditions. As applications, we derive McKean type theorems for the first eigenvalue of Finsler manifolds, as well as generalize to Finsler manifolds a result on fundamental groups due to Milnor.The research of the Y. L. Xin was partially supported by NSFC of and SFECC.     

Local gradient estimate for harmonic functions on Finsler manifolds

In this paper, we prove the local gradient estimate for harmonic functions on complete, noncompact Finsler measure spaces under the condition that the weighted Ricci curvature has a lower bound. As applications, we obtain Liouville type theorems on noncompact Finsler manifolds with nonnegative Ricci curvature.     

Properties of Berwald scalar curvature

We prove that a Finsler manifold with vanishing Berwald scalar curvature has zero E-curvature. As a consequence, Landsberg manifolds with vanishing Berwald scalar curvature are Berwald manifolds. For (α,β)-metrics on manifold of dimension greater than 2, if the mean Landsberg curvature and the Berwald scalar curvature both vanish, then the Berwald curvature also vanishes.     

关于Finsler流形中的距离函数与测地球

利用 Finsler流形中的切曲率和旗曲率 ,研究了距离函数与测地球的凸性 ;指出了在单连通完备 Minkowski空间中测地球正好是平面的一部分     

Finsler interpolation inequalities

We extend Cordero-Erausquin et al.’s Riemannian Borell–Brascamp–Lieb inequality to Finsler manifolds. Among applications, we establish the equivalence between Sturm, Lott and Villani’s curvature-dimension condition and a certain lower Ricci curvature bound. We also prove a new volume comparison theorem for Finsler manifolds which is of independent interest.     

S-curvature of Doubly Warped Product of Finsler Manifolds

Doubly warped product of Finsler manifolds is useful in theoretical physics, particularly in general relativity. In this paper, we study doubly warped product of Finsler manifolds with isotropic mean Berwald curvature or weak isotropic S-curvature.     

Finsler Manifolds with Positive Constant Flag Curvature

It is shown that a Finsler metric with positive constant flag curvature and vanishing mean tangent curvature must be Riemannian. As applications, we also discuss the case of Cheng's maximal diameter theorem and Green's maximal conjugate radius theorem in Finsler manifolds.     

On doubly warped product Finsler manifolds

In this paper, we introduce horizontal and vertical warped product Finsler manifolds. We prove that every C-reducible or proper Berwaldian doubly warped product Finsler manifold is Riemannian. Then, we find the relation between Riemannian curvatures of doubly warped product Finsler manifold and its components, and consider the cases that this manifold is flat or has scalar flag curvature. We define the doubly warped Sasaki-Matsumoto metric for warped product manifolds and find a condition under which the horizontal and vertical tangent bundles are totally geodesic. We obtain some conditions under which a foliated manifold reduces to a Reinhart manifold. Finally, we study an almost complex structure on the tangent bundle of a doubly warped product Finsler manifold.     

Wu's theorem for Kähler–Finsler spaces

In this paper, we study strongly convex Kähler–Finsler manifolds. We prove two theorems: A strongly convex Kähler–Berwald manifold with a pole is a Stein manifold if it has nonpositive horizontal radial flag curvature; A strongly convex Kähler–Finsler manifold with a complex pole is a Stein manifold if it has nonpositive horizontal radial flag curvature.     

Comparison theorems on Riemannian-Finsler manifolds with curvature quartic decay and their applications

We establish some comparison theorems on Finsler manifolds with curvature quartic decay. As their applications, we obtain some optimal compact theorems, volume growth and Mckean type estimate for the first Dirichlet eigenvalue for such manifolds. Although we present the results for Finsler manifolds, they are all new results for Riemannian manifolds.     

关于射影平坦Finsler空间

本文研究了射影平坦Finsler空间的几何量及其几何性质。证明了射影平坦Finsler空间的Ricci曲率可完全由射影因子简洁地刻画出来。同时还证明了，在射影平坦Finsler空间中，平均Berwald曲率S=0意味着Ricci曲率Ric是二次齐次的。此外，给出了一个射影平坦Finsler空间成为常曲率空间或局部Minkowski空间的充分条件。     

Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds with dimension 3

By using the nice behavior of the Hawking mass of the slices of a weak solution of inverse mean curvature flow in three-dimensional asymptotically hyperbolic manifolds, we are able to show that each slice of the flow is star-shaped after a long time, and then we get the regularity of the weak solution of inverse mean curvature flow in asymptotically hyperbolic manifolds. As an application, we prove that the limit of the Hawking mass of the slices of a weak solution of inverse mean curvature flow with any connected C~2-smooth surface as initial data in asymptotically anti-de Sitter-Schwarzschild manifolds with positive mass is greater than or equal to the total mass, which is completely different from the situation in the asymptotically flat case.     

A Schur type lemma for the mean Berwald curvature in Finsler geometry

In this short paper, we study a symmetric covariant tensor in Finsler geometry, which is called the mean Berwald curvature. We first investigate the geometry of the fibres as the submanifolds of the tangent sphere bundle on a Finsler manifold. Then we prove that if the mean Berwald curvature is isotropic along fibres, then the Berwald scalar curvature is constant along fibres.     

A Cartan–Hadamard Theorem for Banach–Finsler Manifolds

In this paper we study Banach–Finsler manifolds endowed with a spray which have seminegative curvature in the sense that the corresponding exponential function has a surjective expansive differential in every point. In this context we generalize the classical theorem of Cartan–Hadamard, saying that the exponential function is a covering map. We apply this to symmetric spaces and thus obtain criteria for Banach–Lie groups with an involution to have a polar decomposition. Typical examples of symmetric Finsler manifolds with seminegative curvature are bounded symmetric domains and symmetric cones endowed with their natural Finsler structure which in general is not Riemannian.     

Finsler空间上的Weyl曲率

The Weyl curvature of a Finsler metric is investigated. This curvature constructed from Riemannain curvature. It is an important projective invariant of Finsler metrics. The author gives the necessary conditions on Weyl curvature for a Finsler metric to be Randers metric and presents examples of Randers metrics with non-scalar curvature. A global rigidity theorem for compact Finsler manifolds satisfying such conditions is proved. It is showed that for such a Finsler manifold,if Ricci scalar is negative,then Finsler metric is of Randers type.     

Metric projections versus non-positive curvature

In this paper two metric properties on geodesic length spaces are introduced by means of the metric projection, studying their validity on Alexandrov and Busemann NPC spaces. In particular, we prove that both properties characterize the non-positivity of the sectional curvature on Riemannian manifolds. Further results are also established on reversible/non-reversible Finsler–Minkowski spaces.     

Hausdorff dimensions of sets related to Lüroth expansion

We study a class of Finsler metrics which contains the class of P-reducible metrics. Finsler metrics in this class are called generalized P-reducible metrics. We consider generalized P-reducible metrics with scalar flag curvature and find a condition under which these metrics reduce to C-reducible metrics. This generalizes Matsumoto’s theorem, which describes the equivalency of C-reducibility and P-reducibility on Finsler manifolds with scalar curvature. Then we show that generalized P-reducible metrics with vanishing stretch curvature are C-reducible.     

Minkowski空间的等价性定理及在Finsler几何的应用

首先利用中心仿射几何中的结果建立了Minkowski空间的等价性定理.作为在Finsler几何中的应用,我们证明满足一定条件的Landsberg空间为Berwald空间,这些条件可以是具有闭的Cartan型形式,S曲率为零或平均Berwald曲率为零.