数学年刊 第27卷B辑第2期(2006)目次和提要

Ricci 曲率和基本群忻元龙在 Ricci 曲率的一定条件下,对体积增长作细化估计,从而推广了 J．Milnor(1968)的二个基本群增长的经典结果． Ricci 孤立子的几何曹怀东 Ricci 孤立子一方面是爱因斯坦(Einstein)度量的自然推广,另一方面也是汉密尔顿(Hamilton) Ricci 流的特殊解,文中介绍了 Ricci 孤立子研究中的一些新进展及其在 Ricci 流的奇异性研究中的     

非塌缩稳态型Ricci孤立子的刚性研究

本文是有关稳态型Ricci孤立子刚性研究的一篇综述报告.特别地,本文讨论了最近作者在非塌缩稳态型Ricci孤立子分类方面的一些工作.     

Camassa-Holm方程凹凸尖峰及光滑孤立子解

研究一类完全可积的新型浅水波方程Camassa-Holm方程的行波孤立子解及双孤立子解.引入凹凸尖峰孤立子及光滑孤立子的概念,研究得到该方程的行波解中具有尖峰性质的凹凸尖峰孤立子解及光滑孤立子解.同时利用Backlund变换给出该类方程的新的双孤立子解.     

非手征对称的共形可积Toda场的孤立子解

用Hirota方法给出了左右不对称的共形可积Toda场的单孤立子解和双孤立子解.     

几类可积的非线性常微分方程(Ⅰ)——一阶方程

本文给出一阶非线性常微分方程的几个可积性结果和几类可积方程,指出许多巳知的可积性结果和可积方程都是它们的特例.这些方程可望在物理学、力学和推导孤立子方程及寻求孤立子解中护到应用.     

关于完备收缩的Ricci-harmonic孤子的研究

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.     

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.     

M.Wadati可积非线性发展方程的精确行波解

用平面动力系统方法研究由M.Wadati提出的一类可积非线性发展方程的精确行波解,获得了该方程的扭波、反扭波解,周期波解和不可数无穷多光滑孤立波解的精确的参数表达式,以及上述解存在的参数条件．     

在$L^p$条件下完备Ricci孤立子流形的刚性定理

我们证明了在一定曲率和$L^p$条件下完备Ricci孤立子流形的一些刚性结果.     

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.     

Generalized quasi-Einstein manifolds with harmonic Weyl tensor

In this paper we introduce the notion of generalized quasi-Einstein manifold that generalizes the concepts of Ricci soliton, Ricci almost soliton and quasi-Einstein manifolds. We prove that a complete generalized quasi-Einstein manifold with harmonic Weyl tensor and with zero radial Weyl curvature is locally a warped product with (n ? 1)-dimensional Einstein fibers. In particular, this implies a local characterization for locally conformally flat gradient Ricci almost solitons, similar to that proved for gradient Ricci solitons.     

Certain Contact Metrics as Ricci Almost Solitons

We show that if a compact K-contact metric is a gradient Ricci almost soliton, then it is isometric to a unit sphere S ²ⁿ⁺¹. Next, we prove that if the metric of a non-Sasakian (κ, μ)-contact metric is a gradient Ricci almost soliton, then in dimension 3 it is flat and in higher dimensions it is locally isometric to E ⁿ⁺¹ ×  S ⁿ (4). Finally, a couple of results on contact metric manifolds whose metric is a Ricci almost soliton and the potential vector field is point wise collinear with the Reeb vector field of the contact metric structure were obtained.     

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.     

Locally Conformally Flat Lorentzian Gradient Ricci Solitons

It is shown that locally conformally flat Lorentzian gradient Ricci solitons are locally isometric to a Robertson–Walker warped product, if the gradient of the potential function is nonnull, and to a plane wave, if the gradient of the potential function is null. The latter gradient Ricci solitons are necessarily steady.     

Semiumbilic points for minimal surfaces in Euclidean 4-space

We show that locally conformally flat gradient Ricci solitons, possibly incomplete, are locally isometric to a warped product of an interval and a space form. Consequently, we get that complete gradient shrinking and steady Ricci solitons with vanishing Weyl tensor are rotationally symmetric, from which their classification follows.     

Bach-flat gradient steady Ricci solitons

In this paper we prove that any n-dimensional (n ≥ 4) complete Bach-flat gradient steady Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton. We also show that a three-dimensional gradient steady Ricci soliton with divergence-free Bach tensor is either flat or isometric to the Bryant soliton. In particular, these results improve the corresponding classification theorems for complete locally conformally flat gradient steady Ricci solitons in Cao and Chen (Trans Am Math Soc 364:2377–2391, 2012) and Catino and Mantegazza (Ann Inst Fourier 61(4):1407–1435, 2011).     

Hitchin–Thorpe inequality and Kaehler metrics for compact almost Ricci soliton

The purpose of this paper is to prove a Hitchin–Thorpe inequality for a four-dimensional compact almost Ricci soliton. Moreover, we prove that under a suitable integral condition, a four-dimensional compact almost Ricci soliton is isometric to standard sphere. Finally, we prove that under a simple condition, a four-dimensional compact Ricci soliton with harmonic self-dual part of Weyl tensor is either isometric to a standard sphere \(\mathbb S ^{4}\) or is Kaehler–Einstein.     

Compact almost Ricci solitons with constant scalar curvature are gradient

The aim of this note is to prove that any compact non-trivial almost Ricci soliton $\big (M^n,\,g,\,X,\,\lambda \big )$ with constant scalar curvature is isometric to a Euclidean sphere $\mathbb {S}^{n}$ . As a consequence we obtain that every compact non-trivial almost Ricci soliton with constant scalar curvature is gradient. Moreover, the vector field $X$ decomposes as the sum of a Killing vector field $Y$ and the gradient of a suitable function.     

Steady Ricci solitons with horizontally ε-pinched Ricci curvature

In this paper,we prove that any κ-noncollapsed gradient steady Ricci soliton with nonnegative curvature operator and horizontally κ-pinched Ricci curvature must be rotationally symmetric.As an application,we show that any κ-noncollapsed gradient steady Ricci soliton(Mn,g,f) with nonnegative curvature operator must be rotationally symmetric if it admits a unique equilibrium point and its scalar curvature R(x) satisfies lim_(ρ(x)→∞) R(x)f(x)=C_0 sup_(x∈M) R(x) with C_0n-2/2.