Limit sets of geometrically finite groups acting on Busemann spaces

In this paper, we investigate limit sets of geometrically finite groups acting on Busemann spaces. We show a Busemann space analogue of several results proved by A. Ranjbar-Motlagh for geometrically finite groups acting on hyperbolic spaces in the sense of Gromov.     

Busemann Functions and Barrier Functions

On a smooth, non-compact, complete, boundaryless, connected Riemannian manifold there are two kinds of functions: Busemann functions with respect to rays and barrier functions with respect to lines (if there exists at least one). In this paper we collect some known properties on Busemann functions and introduce some new fundamental properties on barrier functions. Based on these properties of barrier functions, we could define some relations on the set of lines and thus classify them. With the equivalence relation we introduced, we present a generalization of a rigidity conjecture.     

The Busemann theorem for complex p-convex bodies

The Busemann theorem states that the intersection body of an origin-symmetric convex body is also convex. In this paper, we prove a version of the Busemann theorem for complex p-convex bodies. Namely that the complex intersection body of an origin-symmetric complex p-convex body is γ-convex for certain γ. The result is the complex analogue of the work of Kim, Yaskin, and Zvavitch on (real) p-convex bodies. Furthermore, we show that the generalized radial qth mean body of a p-convex body is γ-convex for certain γ.     

Action potential and weak KAM solutions

For convex superlinear lagrangians on a compact manifold M we characterize the Peierls barrier and the weak KAM solutions of the Hamilton-Jacobi equation, as defined by A. Fathi [9], in terms of their values at each static class and the action potential defined by R. Ma né [14]. When the manifold M is non-compact, we construct weak KAM solutions similarly to Busemann functions in riemannian geometry. We construct a compactification of by extending the Aubry set using these Busemann weak KAM solutions and characterize the set of weak KAM solutions using this extended Aubry set. Received: 13 November 2000 / Accepted: 4 December 2000 / Published online: 25 June 2001     

Critical phenomenon for a percolation model

We consider a percolation model which consists of oriented lines placed randomly on the plane. The lines are of random length and at a random angle with respect to the horizontal axis and are placed according to a Poisson point process; the length, angle, and orientation being independent of the underlying Poisson process. We establish a critical behaviour of this model, i.e., percolation occurs for large intensity of the Poisson process and does not occur for smaller intensities. In the special case when the lines are of fixed unit length and are either oriented vertically up or oriented horizontally to the left, with probability p or (1-p), respectively, we obtain a lower bound on the critical intensity of percolation.     

第二类典型域的Busemann函数

本文给出了第二类典型城的 Ｂｕｓｅｍａｎｎ函数．证明了下面的定理：对 Ｒ-ＩＩ（ｐ）中通过０点的任一测地线其中Ｕ为 ｐ阶酉方阵,．则由ｒ决定的 Ｂｕｓｅｍａｎｎ函数为：．其中 为下列矩阵的对角元素：     

Baxter-Guttmann-Jensen conjecture for power series in directed percolation problem

A probabilistic model of a flow of fluid through a random medium,percolation model, provides a typical example of statistical mechanical problems which are easy to describe but difficult to solve. While the percolation problem on undirected planar lattices is exactly solved as a limit of the Potts models, there still has been no exact solution for the directed lattices. The most reliable method to provide good approximations is a numerical estimation using finite power-series expansion data of the infinite formal power series for percolation probability. In order to calculate higher-order terms in power series, Baxter and Guttmann [6] and Jensen and Guttmann [33] proposed an extrapolation procedure based on an assumption that thecorrection terms, which show the difference between the exact infinite power series and approximate finite series, are expressed as linear combinations of the Catalan numbers.In this paper, starting from a brief review on the directed percolation problem and the observation by Baxter, Guttmann, and Jensen, we state some theorems in which we explain the reason why the combinatorial numbers appear in the correction terms of power series. In the proof of our theorems, we show several useful combinatorial identities for the ballot numbers, which become the Catalan numbers in a special case. These identities ensure that a summation of products of the ballot numbers with polynomial coefficients can be expanded using the ballot numbers. There is still a gap between our theorems and the Baxter-Guttmann-Jensen observation, and we also give some conjectures.As a generalization of the percolation problem on a directed planar lattice, we present two topics at the end of this paper: The friendly walker problem and the stochastic cellular automata in higher dimensions. We hope that these two topics as well as the directed percolation problem will be of much interest to researchers of combinatorics.     

Geometry of ideal boundaries of geodesic spaces with nonpositive curvature in the sense of Busemann

We establish relations between different approaches to the ideal closure of a geodesic metric space with nonpositive curvature in the sense of Busemann. We construct the counterexample showing that the Busemann ideal closure can differ from the geodesic closure.     

Solution of a Busemann problem

We give a solution to the problem posed by Busemann which is as follows: Determine the noncompact Busemann G-spaces such that for every two geodesics there exists a motion taking one to the other. We prove that each of these spaces is isometric to the Euclidean space or to one of the noncompact symmetric spaces of rank 1 (of negative sectional curvature).     

Locally G-homogeneous Busemann G-spaces

We present short proofs of all known topological properties of general Busemann G-spaces (at present no other property is known for dimensions more than four). We prove that all small metric spheres in locally G-homogeneous Busemann G-spaces are homeomorphic and strongly topologically homogeneous. This is a key result in the context of the classical Busemann conjecture concerning the characterization of topological manifolds, which asserts that every n-dimensional Busemann G-space is a topological n-manifold. We also prove that every Busemann G-space which is uniformly locally G-homogeneous on an orbal subset must be finite-dimensional.     

Some applications of cross-section measures in Minkowski spaces

Summary The purpose of this paper is to establish extremal values for inner and outer radii of the unit ball of a Minkowski space for the Holmes--Thompson and Busemann measures. Furthermore, we confirm a conjecture of C. A. Rogers and G. C. Shephard on ellipsoids.     

Comparing volumes by concurrent cross-sections of complex lines: a Busemann–Petty type problem

We consider the problem of comparing the volumes of two star bodies in an even-dimensional Euclidean space \({\mathbb {R}}^{2n} = {\mathbb {C}}^n\) by comparing their cross sectional areas along complex lines (special 2-dimensional real planes) through the origin. Under mild symmetry conditions on one of the bodies a Busemann–Petty type theorem holds. Quaternionic and octonionic analogs also hold. The argument relies on integration in polar coordinates coupled with Jensen’s inequality. Along the way we provide a criterion that detects which centered bodies are circular. i.e., stabilized by multiplication by complex numbers of unit modulus. Our goal is to present a Busemann–Petty type result with a minimum of required background (in the spirit of L. K. Hua’s book on the classical domains) and, in addition, to suggest characterizations of classes of star bodies by means of integral geometric inequalities.     

Boundaries of non-compact harmonic manifolds

In this paper we consider non-compact non-flat simply connected harmonic manifolds. In particular, we show that the Martin boundary and Busemann boundary coincide for such manifolds. For any finite volume quotient we show that (up to scaling) there is a unique Patterson–Sullivan measure and this measure coincides with the harmonic measure. As an application of these results we prove that the geodesic flow on a non-flat finite volume harmonic manifold without conjugate points is topologically transitive.     

Second-order metropolitan urban phase transitions

The morphology evolution of Metropolitan Urban Areas constituted by different Central Business Districts is studied in this paper. For this matter, we propose a stochastic model which combines an initial percolation setting followed by a diffusion-limited aggregation mechanism. Our model mimics better than either case (percolation or diffusion-limited aggregation) the Metropolitan Urban Areas formation progress. We argue that the Metropolitan Urban Areas case introduced in this paper, grows in such a way that undergoes a non-equilibrium second-order phase transition during this process. This conclusion is supported by a fractal dimension and configurational entropy analysis, as well as by studying an empirical case.     

对称空间SU~*(2n)/sp(n)上的Busemann函数

Busemann函数在对完备Riemann流形上的拓扑与几何问题的研究中起作十分重要的作用。而显式求出一个域上的Busemann函数将有助于一些问题的解决。在本文中,我们显式求出了对称空间SU~*(2n)/sp(n)上的Busemann函数。     

Concentration results for a Brownian directed percolation problem

We consider the hydrodynamic limit for a certain Brownian directed percolation model, and establish uniform concentration results. In view of recent work on the connection between this directed percolation model and eigenvalues of random matrices, our results can also be interpreted as uniform concentration results at the process level for the largest eigenvalue of Hermitian Brownian motion.     

Approximating fixed points of mappings satisfying condition (E) in Busemann space

It is well-known that in a Banach space, using the Ishikawa iterative process, one can find fixed points of nonexpansive mappings via asymptotic center’s method. In this paper, we obtain the fixed points of mappings satisfying so-called condition (E) in a uniformly convex Busemann space. Many known results in CAT (0) spaces are improved and extended by our results.     

Busemann Functions in a Harmonic Manifold

For a noncompact complete and simply connected harmonic manifold M, we prove the analyticity of Busemann functionson M. This is the main result of this paper. An application of it shows that the harmonic spaces having minimal horospheres have the bi-asymptotic property. Finally, we prove that the total Busemann functionis continuous in C topology. As a consequence, we show that the uniform divergence of geodesics holds in these spaces.     

On dual quermassintegrals of mixed intersection bodies

In this paper, it is shown that a family of inequalities involving mixed intersection bodies holds. The Busemann intersection inequality is the first of this family. All of the members of this family are strengthened versions of classical inequalities between pairs of dual quermassintegrals of a star body.     

关于H. Wu问题

著名几何学家Ｈ．Ｗｕ在「４」中提出这样的问题：若一完备非紧的黎曼流形仅有两个符号相反的Ｂｕｓｅｍａｎｎ函数,则该流形的结构如何？本文在流形具非负Ｒｉｃｃｉ曲率或截曲率具下界的情况下部分地解决了这一问题。同时还讨论了流形在具非负曲率条件下有关的一些性质。