关于完备收缩的Ricci-harmonic孤子的研究（英文）

本文研究了收缩的Ricci-harmonic孤子的几何性质的问题.利用文献[4]在Ricci孤子下的方法,获得了每个紧致Ricci-harmonic孤子是一个梯度孤子的结论,推广了Perelman等人在Ricci孤子下的结果.此外,利用文献[14]在Ricci孤子下的方法,获得了完备非紧梯度收缩的Ricci-harmonic孤子具有比至多欧氏增长更加精确的体积增长估计的结果,推广了文献[14]在Ricci孤子下的结果.     

Finsler流形上的Laplace比较定理和有界估计(英文)

本文研究了Finsler流形上的距离函数的Laplacian.利用指标引理和文献[4]中主要方法,获得了Ricci曲率有函数下界的Laplacian比较定理,改进了文献[6]和文献[7]的相关结果.     

光滑度量测度空间上的加权扩散方程的梯度估计

本文主要考虑了一类加权非线性扩散方程正解的梯度估计.在m-维Bakry-(E)mery Ricci曲率下有界的假设下,得到加权多孔介质方程(γ＞1)正解的Li-Yau型梯度估计,此外对于加权快速扩散方程(0＜γ＜1),证明了Hamilton型椭圆梯度估计,结论分别推广了Lu,Ni,Vázquez and Villani在文[1]和Zhu在文[2]中的结果.     

关于带有Ricci孤子的trans-Sasakian流形的注记(英文)

本文主要研究带有Ricci孤子的(α,β)型trans-Sasakian流形,证明了带有Ricci孤子(g,ξ,λ)的3-维紧致trans-Sasakian流形是一个Sasakian流形.此外,如果α,β是常数,得到带有梯度Ricci孤子的trans-Sasakian流形是Einstein流形.     

一类扭积形式的梯度近Ricci孤立子

研究了扭积和梯度近Ricci孤立子的关系问题.获得了一类扭积形式的梯度近Ricci孤立子,推广了梯度近Ricci孤立子的存在范围.     

Robin边界条件下的抛物方程爆破分析

本文研究了Robin边界条件下含有梯度项的一类抛物方程解的爆破问题.利用合适的Sobolve型与一阶微分不等式等方法,获得了当方程的解发生爆破时其爆破时间的下界估计与解不发生爆破的条件.推广了文献[4]的结果.     

关于一个Calabi-型曲率流的局部存在性

基于文[14]的讨论,本文将针对一个紧致无边黎曼流形上关于Ricci曲率的L2-模的负梯度流这一4阶退化抛物型方程组,首先给出其解的局部存在性的详细证明,其次,将在文[14]结果的基础上,进一步在关于此流的奇异性方面讨论解的另一类爆破性态.     

非线性SCHR?DINGER-MAXWELL方程的基态解

本文研究了schr?dinger-Maxwell方程基态解存在性的问题.在V,K,f,g满足文中定理1.1的假设条件下,利用山路定理的方法,获得了系统(NSM)的基态解这一结果,推广了文献[1]中0 p 1和文献[2]中系统高能解的结果.     

Hardy-Hilbert不等式的推广

本文研究了Hardy-Hilbert不等式的推广.利用β函数,获得了Hardy-Hilbert不等式推广的一种统一模式,且推广了杨必成在文献[2,4]中的已知结果.     

某种一般非参数回归函数估计的强相合

本文研究了较为一般的非参数回归函数估计的问题.利用传输不等式,获得了参数估计的强相合性,推广了文献[1]、[6]等结果.     

The conjugate heat equation and Ancient solutions of the Ricci flow

We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of Type I κ-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman?s previous result on backward limits of κ-solutions in dimension 3, in which case the curvature operator is nonnegative (it follows from Hamilton–Ivey curvature pinching estimate). As an application, this also addresses an issue left in Naber (2010) [23], where Naber proves the interesting result that there exists a Type I dilation limit that converges to a gradient shrinking Ricci soliton, but that soliton might be flat. The Gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest.     

On the essential spectrum of complete non-compact manifolds

In this paper, we prove that the Lp essential spectra of the Laplacian on functions are [0,+∞) on a non-compact complete Riemannian manifold with non-negative Ricci curvature at infinity. The similar method applies to gradient shrinking Ricci soliton, which is similar to non-compact manifold with non-negative Ricci curvature in many ways.     

Volume growth for gradient shrinking solitons of Ricci-harmonic flow

In this paper,we derive an estimate on the potential functions of complete noncompact gradient shrinking solitons of Ricci-harmonic flow,and show that complete noncompact gradient shrinking Ricci-harmonic solitons have Euclidean volume growth at most.     

Gap theorems for Kähler-Ricci solitons

In this paper, we prove that a gradient shrinking compact K?hler-Ricci soliton cannot have too large Ricci curvature unless it is K?hler-Einstein. Received: 23 October 2007, Revised: 28 February 2008     

Compact gradient shrinking Ricci solitons with positive curvature operator

In this article, we first derive several identities on a compact shrinking Ricci soliton. We then show that a compact gradient shrinking soliton must be Einstein, if it admits a Riemannian metric with positive curvature operator and satisfies an integral inequality. Furthermore, such a soliton must be of constant curvature.     

Rigidity of shrinking Ricci solitons

We show that a compact Ricci soliton is rigid if and only if the Weyl conformal tensor is harmonic. In the complete noncompact case we prove the same result assuming that the curvature tensor has at most exponential growth and the Ricci tensor is bounded from below.     

On the Potential Function of Gradient Steady Ricci Solitons

In this paper, we study the potential function of gradient steady Ricci solitons. We prove that the infimum of the potential function decays linearly. As a consequence, we show that a gradient steady Ricci soliton with bounded potential function must be trivial, and that no gradient steady Ricci soliton admits uniformly positive scalar curvature.     

On <Emphasis Type="Italic">f</Emphasis>-non-parabolic ends for Ricci-harmonic metrics

In this paper, we study gradient Ricci-harmonic soliton metrics and quasi Ricci-harmonic metrics (both metrics are called Ricci-harmonic). First, we prove that all ends of \(\tau \)-quasi Ricci-harmonic metrics with \(\tau >1\) should be f-non-parabolic if \(\lambda =0,\mu >0\), or \(\lambda <0, \mu \ge 0\). For the case that \(\lambda<0, \mu < 0\), we can also arrive at the f-non-parabolic property if we add a condition about the scalar curvature. Furthermore, we discuss the connectivity at infinity for quasi Ricci-harmonic metrics. We also conclude that all ends of steady or expanding gradient Ricci-harmonic solitons should be f-non-parabolic, based on which we establish structure theorems for these two solitons.     

A compactness result for Kähler Ricci solitons

In this paper we prove a compactness result for compact Kähler Ricci gradient shrinking solitons. If (Mi,gi) is a sequence of Kähler Ricci solitons of real dimension n?4, whose curvatures have uniformly bounded Ln/2 norms, whose Ricci curvatures are uniformly bounded from below and μ(gi,1/2)?A (where μ is Perelman's functional), there is a subsequence (Mi,gi) converging to a compact orbifold (M_∞,g_∞) with finitely many isolated singularities, where g_∞ is a Kähler Ricci soliton metric in an orbifold sense (satisfies a soliton equation away from singular points and smoothly extends in some gauge to a metric satisfying Kähler Ricci soliton equation in a lifting around singular points).     

On Kropina metrics of scalar flag curvature

We classify Kropina metrics of weakly isotropic flag curvature in dimension greater than two. Moreover, we prove that every Einstein Kropina metric in dimension greater than two is a Ricci constant metric with vanishing S-curvature in different way from Zhang and Shen (2013) [14] and prove the three-dimensional rigidity theorem for an Einstein Kropina metric.