Ricci孤立子的几何分析性质及其分类问题研究

As an analytic method, curvature flow has played an important role in geometric problems. It has become a hotspot in the area of geometry. As the self-similar solution to the equation of curvature flow and the geometric model of singularity, the soliton has significant value in analyzing the properties of the curvature flow and understanding the geometry of singularity. Meanwhile, the soliton itself is an important geometric structure. It has great theoretical value. The essence of the research in solitons is to solve the problem of classification. Comparing to the compact case in which we already have plenty of tools and results, the project focuses on the classification problem of solitons in complete non-compact case. For shrinking gradient Ricci soliton, we study its classification problem under the condition of Ricci curvature or harmonic Bach tensor. For 4-dimensional steady gradient Ricci soliton, we search for the lower bound of its curvature operator and study its asymptotic scalar curvature ratio. Under the condition of half-harmonic Weyl tensor, we study the classification of 4-dimensional steady gradient Ricci soliton. By loosing the premise of current studies, the project can describe the geometric and topological structure of solitons under certain curvature conditions more precisely. It has important applications in the field of Ricci flow.

曲率流作为分析方法在几何问题中发挥了重大作用，是几何领域的研究热点。作为曲率流的自相似解和奇点处的几何模型，孤立子对分析曲率流性质和理解奇点的几何具有重要意义。同时孤立子也是特殊的几何结构，具有重要的研究价值。孤立子研究的实质是解决其分类问题。紧致情况的孤立子分类已具备较丰富的方法和研究基础，本项目主要研究完备非紧情况Ricci流孤立子的分类问题。对收缩梯度Ricci孤立子，分别研究其Ricci曲率条件和调和Bach张量条件下的分类问题；对四维稳恒梯度Ricci孤立子，研究其曲率算子的下界与渐进数量曲率比；在半调和Weyl张量条件下研究四维稳恒梯度Ricci孤立子的分类问题。本项目减弱了现有研究基础的前提限制，能够更好地描述在一定曲率条件下孤立子的几何拓扑结构，在Ricci流领域有重要应用。

曲率流作为分析方法在几何问题中发挥了重大作用，是几何领域的研究热点。作为曲率流的自相似解和奇点处的几何模型，孤立子对分析曲率流性质和理解奇点的几何具有重要意义。同时孤立子也是特殊的几何结构，具有重要的研究价值。孤立子研究的实质是解决其分类问题。本项目主要研究四维及一般维梯度Ricci孤立子的分类问题、梯度Kähler-Ricci孤立子的分类问题以及流形上二阶抛物型偏微分方程的梯度估计。得到了如黎曼曲率的四阶散度为零的一般维梯度收缩Ricci孤立子是刚性的、B-张量为零的紧致梯度Kähler-Ricci孤立子是Kähler-Einstein以及Ricci流下非线性抛物型方程的局部椭圆型梯度估计等一系列重要结果。本项目减弱了现有研究基础的前提限制，能够更好地描述在一定曲率条件下孤立子的几何拓扑结构，在Ricci流领域有重要应用。