The Valiron-Titchmarsh theorem on asymptotic behavior of entire functions with negative zeros has been recently generalized onto subharmonic functions with the Riesz measure on a half-line in Rn, n≥3. Here we extend ...The Valiron-Titchmarsh theorem on asymptotic behavior of entire functions with negative zeros has been recently generalized onto subharmonic functions with the Riesz measure on a half-line in Rn, n≥3. Here we extend the Drasin complement to the Valiron-Titchmarsh theorem and show that if u is a subharmonic function of this class and of order 0〈ρ〈1, then the existence of the limit limr→∞logu（r）/N（r）, where N（r） is the integrated counting function of the masses of u, implies the regular asymptotic behavior for both u and its associated measure.展开更多
In this paper, we study the infinity behavior of the bounded subharmonic functions on a Ricci non-negative Riemannian manifold M. We first show that if h is a bounded subharmonic function. If we further assume that t...In this paper, we study the infinity behavior of the bounded subharmonic functions on a Ricci non-negative Riemannian manifold M. We first show that if h is a bounded subharmonic function. If we further assume that the Laplacian decays pointwisely faster than quadratically we show that h approaches its supremun pointwisely at infinity, under certain auxiliary conditions on the volume growth of M. In particular, our result applies to the case when the Riemannian manifold has maximum volume growth. We also derive a representation formula in our paper, from which one can easily derive Yau’s Liouville theorem on bounded harmonic functions.展开更多
We prove a Harnack inequality for positive harmonic functions on graphs whichis similar to a classical result of Yau on Riemannian manifolds. Also, we prove a mean valueinequality of nonnegative subharmonic functions ...We prove a Harnack inequality for positive harmonic functions on graphs whichis similar to a classical result of Yau on Riemannian manifolds. Also, we prove a mean valueinequality of nonnegative subharmonic functions on graphs.展开更多
The authors study complete open manifolds whose curvature is non-negative along ray directions. They prove that such manifold has infinite volume. Cheeger-Gromoll's splitting; theorem is generalized. They also stu...The authors study complete open manifolds whose curvature is non-negative along ray directions. They prove that such manifold has infinite volume. Cheeger-Gromoll's splitting; theorem is generalized. They also study topology of such manifolds.展开更多
A complete manifold is said to be nonparabolic if it does admit a positive Green’s function. To find a sharp geometric criterion for the parabolicity/nonparbolicity is an attractive question inside the function theor...A complete manifold is said to be nonparabolic if it does admit a positive Green’s function. To find a sharp geometric criterion for the parabolicity/nonparbolicity is an attractive question inside the function theory on Riemannian manifolds. This paper devotes to proving a criterion for nonparabolicity of a complete manifold weakened by the Ricci curvature. For this purpose, we shall apply the new Laplacian comparison theorem established by the first author to show the existence of a non-constant bounded subharmonic function.展开更多
文摘The Valiron-Titchmarsh theorem on asymptotic behavior of entire functions with negative zeros has been recently generalized onto subharmonic functions with the Riesz measure on a half-line in Rn, n≥3. Here we extend the Drasin complement to the Valiron-Titchmarsh theorem and show that if u is a subharmonic function of this class and of order 0〈ρ〈1, then the existence of the limit limr→∞logu（r）/N（r）, where N（r） is the integrated counting function of the masses of u, implies the regular asymptotic behavior for both u and its associated measure.
文摘In this paper, we study the infinity behavior of the bounded subharmonic functions on a Ricci non-negative Riemannian manifold M. We first show that if h is a bounded subharmonic function. If we further assume that the Laplacian decays pointwisely faster than quadratically we show that h approaches its supremun pointwisely at infinity, under certain auxiliary conditions on the volume growth of M. In particular, our result applies to the case when the Riemannian manifold has maximum volume growth. We also derive a representation formula in our paper, from which one can easily derive Yau’s Liouville theorem on bounded harmonic functions.
基金supported by the National Science Foundation of China(11671401)
文摘We prove a Harnack inequality for positive harmonic functions on graphs whichis similar to a classical result of Yau on Riemannian manifolds. Also, we prove a mean valueinequality of nonnegative subharmonic functions on graphs.
文摘The authors study complete open manifolds whose curvature is non-negative along ray directions. They prove that such manifold has infinite volume. Cheeger-Gromoll's splitting; theorem is generalized. They also study topology of such manifolds.
基金supported by the National Natural Science Foundation of China(Nos.12071080,12141104)。
文摘A complete manifold is said to be nonparabolic if it does admit a positive Green’s function. To find a sharp geometric criterion for the parabolicity/nonparbolicity is an attractive question inside the function theory on Riemannian manifolds. This paper devotes to proving a criterion for nonparabolicity of a complete manifold weakened by the Ricci curvature. For this purpose, we shall apply the new Laplacian comparison theorem established by the first author to show the existence of a non-constant bounded subharmonic function.
