Ricci孤子的几何分析性质

We focus on geometric properties of the soliton of compelete Riemannian manifolds under curvature flows.. Firstly, Ricci flow and mean curvature flow have been proved to be important tools in solving geometric problems. Based on the results that mathematicians have got recently, we consider further geometric and topologic properties of the soliton of Riemannian manifolds. For example, we hope that we can get better estimatians to potential functions of gradient Ricci solitons.We also hope to find the relationship between the condition "Bach flat" and the geometric properties of Ricci solitons.. Secondly, the problems of the mean curvature flow in higher codimension are much harder than the case of hypersurfaces. We wish to find suitable conditions under which the properties of the soliton in codimension-1 case can be generalized to the case of higher codimension. Go a step further, mean curvature flow has been proved to be an effect tool in the field of the geometry of sub-manifolds. As an application of our results, we wish to consider corresponding problems in the field of sub-manifolds.. Finally, there are several curvature flows which plays an important role in other fields. We will also discuss the geometric properties of the soliton of Yamabe flow and Ricci-harmonic flow. . These are important and siganificative problems in geometric analysis.

本项目主要研究在Ricci流下，完备黎曼流形孤子的几何性质。. 首先，在近年来的进展上研究梯度Ricci孤子的进一步的几何拓扑性质。针对完备非紧的情况，讨论对Ricci孤子的势函数更好的估计，研究Bach flat条件与孤子几何性质的关系。其次，对于高余维数的平均曲率流，我们将对其孤子的性质进行系统研究，希望在适当的条件下，其孤子有类似于超曲面情形或Ricci孤子的几何拓扑性质，并且能用平均曲率流来考虑子流形几何的相关问题。最后，对重要的曲率流，如Yamabe流、Ricci-harmonic流等，我们也将对它们的孤子的情况进行深入讨论。. 这是几何分析领域重要而有意义的课题。

本项目在近年来的研究进展上进一步研究了Ricci孤子的几何性质. 本项目对Ricci 孤子的研究重点放在了对Ricci孤子的一般情况：一般化的quasi-Einstein 度量进行研究. 对于Yamabe 孤子和Ricci-harmonic孤，本项目也进行了深入研究. 具体取得了如下成果.. 首先，本项目证明了一般化的quasi-Einstein度量下Perelman的v-熵的二阶变分公式. 其次，本项目证明了每个紧致收缩Ricci-harmonic孤子是梯度的结论，并且对完备非紧梯度收缩Ricci-harmonic孤子给出了更加精确的体积增长估计. 第三，本项目对调和Weyl 张量下的紧致和完备非紧Yamabe孤子做了归类. 最后，本项目证明了一般的quasi-Einstein度量下的Myers定理.