Liouville property for groups and manifolds

We introduce a method to estimate the entropy of random walks on groups. We apply this method to exhibit the existence of compact manifolds with amenable fundamental groups such that the universal cover is not Liouville. We also use the criterion to prove that a finitely generated solvable group admits a symmetric measure with non-trivial Poisson boundary if and only if this group is not virtually nilpotent. This, in particular, shows that any polycyclic group admits a symmetric measure such that its boundary does not readily interprete in terms of the ambient Lie group. As another application we get a series of examples of amenable groups such that any finite entropy non-degenerate measure on them has non-trivial Poisson boundary. Since the groups in question are amenable, they do admit measures such that the corresponding random walks have trivial boundary; the above shows that such measures on these groups have infinite entropy. Mathematics Subject Classification (1991) 60B15, 60J50, 28D20, 20P05, 43A07, 60J65, 43A85, 20f16     

ON POSITIVE SOLUTIONS OF ONE CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS

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Every planar graph with the Liouville property is amenable

We introduce a strengthening of the notion of transience for planar maps in order to relax the standard condition of bounded degree appearing in various results, in particular, the existence of Dirichlet harmonic function s proved by Benjamini and Schramm. As a corollary we obtain that every planar nonamenable graph admits nonconstant Dirichlet harmonic function s27.     

Liouville theorem and coupling on negatively curved Riemannian manifolds

By using probabilistic approaches, Liouville theorems are proved for a class of Riemannian manifolds with Ricci curvatures bounded below by a negative function. Indeed, for these manifolds we prove that all harmonic functions (maps) with certain growth are constant. In particular, the well-known Liouville theorem due to Cheng for sublinear harmonic functions (maps) is generalized. Moreover, our results imply the Brownian coupling property for a class of negatively curved Riemannian manifolds. This leads to a negative answer to a question of Kendall concerning the Brownian coupling property.     

The Li-Yau-Hamilton estimate and the Yang-Mills heat equation on manifolds with boundary

The paper pursues two connected goals. Firstly, we establish the Li-Yau-Hamilton estimate for the heat equation on a manifold M with nonempty boundary. Results of this kind are typically used to prove monotonicity formulas related to geometric flows. Secondly, we establish bounds for a solution ∇(t) of the Yang-Mills heat equation in a vector bundle over M. The Li-Yau-Hamilton estimate is utilized in the proofs. Our results imply that the curvature of ∇(t) does not blow up if the dimension of M is less than 4 or if the initial energy of ∇(t) is sufficiently small.     

On the matching polynomial of subdivision graphs

Let G be a simple graph and let S(G) be the subdivision graph of G, which is obtained from G by replacing each edge of G by a path of length two. In this paper, by the Principle of Inclusion and Exclusion we express the matching polynomial and Hosoya index of S(G) in terms of the matchings of G. Particularly, if G is a regular graph or a semi-regular bipartite graph, then the closed formulae of the matching polynomial and Hosoya index of S(G) are obtained. As an application, we prove a combinatorial identity.     

Large volume growth and the topology of open manifolds

In this paper, we study complete noncompact Riemannian manifolds with nonnegative Ricci curvature and large volume growth. We find some reasonable conditions to insure that this kind of manifolds are diffeomorphic to a Euclidean space or have finite topological type. Received: January 4, 2000; in final form: October 31, 2000 / Published online: 19 October 2001     

Heisenberg-Pauli-Weyl uncertainty inequalities and polynomial volume growth

In its simpler form, the Heisenberg-Pauli-Weyl inequality says that

     

On the volume functional of compact manifolds with harmonic Weyl tensor

The main aim of this article is to give the complete classification of critical metrics of the volume functional on a compact manifold M with boundary $\partial M$ and under the harmonic Weyl tensor condition. In particular, we prove that a critical metric with a harmonic Weyl tensor on a simply connected compact manifold with the boundary isometric to a standard sphere ${\mathbb{S}}^{n-1}$ must be isometric to a geodesic ball in a simply connected space form ${\mathbb{R}}^n$, ${\mathbb{H}}^n$, and ${\mathbb{S}}^n$. To this end, we first conclude the classification of such critical metrics under the Bach-flat assumption and then we prove that both geometric conditions are equivalent in this situation.     

A Theorem of Liouville Type for Harmonic Morphisms

Let : M N be a harmonic morphism from a complete noncompact Riemannian manifold M with nonnegative Ricci curvature to a complete Riemannian manifold N with nonpositive scalar curvature. We show if the energy of is finite, then is constant. This can be compared with a similar result for harmonic maps when N has nonpositive sectional curvature due to Schoen and Yau.     

一类非线性方程的Dirichlet问题

设D为R~d(d≥3)中非空规则开子集.本文利用超布朗运动给出了一类非线性方程Dirichlet问题非负有界解的通式.在一定条件下证明了满足在无穷远处有规定极限非负有界解的存在唯一性,并给出了解的概率表达式,推广了Dynkin中的结果.     

Distances on Lie groups of polynomial volume growth

For any simply connected solvable Lie group Q of polynomial volume growth, we introduce the notion of nil-shadow of Q. We shall give an explicit formula for the distance to the origin of an element q ∈ Q in terms of its exponential coordinates of the second kind taken in an appropriate basis. This result extends a previous result for nilpotent Lie groups [6, Theorem DN] and [7, Theorem 1].     

Products of constant curvature spaces with a Brownian independence property

The time and place Brownian motion on the product of constant curvature spaces first exits a normal ball of radius centered at the starting point of the Brownian motion are considered. The asymptotic expansions, as decreases to zero, for joint moments of the first exit time and place random variables are computed with error . It is shown that the first exit time and place are independent random variables only if each factor space is locally flat or of dimension three.

     

利用Ito公式求布朗运动和几何布朗运动的矩

利用Ito公式及Ito积分的性质求出了布朗运动和几何布朗运动的矩的一般形式,同时指出可以利用这种方法求其他扩散过程的矩.     

New exact solutions to the nonautonomous Liouville equation

We obtain new formulas for the exact analytic solutions to the nonautonomous elliptic Liouville equation in the two-dimensional coordinate space with the free function dependent specially on an arbitrary harmonic function. We present new exact solutions to the wave Liouville equation with two arbitrary functions, providing original formulas for the general solution for the classical (autonomous) and wave Liouville equations. Some equivalence transformations are presented for the elliptic Liouville equation depending on conjugate harmonic functions. In particular, we indicate a transformation that reduces the equation under study to an autonomous form.