瑞奇调和流与瑞奇调和孤立子的几何分析性质研究

几何分析是非常活跃的研究领域，在过去的几十年里许多重要的几何拓扑问题得到解决，比如庞加莱猜想和球定理。本项目研究瑞奇调和流与瑞奇调和孤立子的几何分析性质。1. 关于瑞奇调和流，假设数量曲率有界，我们考虑其距离函数的扭曲估计，构造出时间导数与 Lapacian 有界的截断函数，进而得到共轭热方程的平均值不等式，最后导出共轭热核的高斯型上下界估计。另一方面，同样假设数量曲率有界，我们证明瑞奇调和流的 Backward Pseudolocality 性质，结合多尺度的数学归纳法，得到黎曼曲率张量积分模的一致控制。这可用来研究瑞奇调和流的渐近行为与延拓，与数量曲率有限时间爆破猜想密切相关。2. 我们考虑瑞奇调和孤立子。一方面我们研究它的数量曲率的衰减估计以及正曲率条件下的分类，另一方面我们研究它在不同曲率条件下的紧性。这对研究瑞奇调和孤立子在无穷远处的渐近行为与分类有重要意义。

Geometric analysis is an acitve field, and many important problems are solved using geometric flow in the past several decades, such as Poincare conjecture and Sphere Theorem. This project is concerned with the geometric and analytic properties of Ricci harmonic flow and Ricci harmonic solition. 1. About the Ricci harmonic flow, on one hand, assuming the boundedness of the scalar curvature, we consider the distance distortion estimate, construct a cutoff function whose time derivative and Laplacian are bounded, then we obtain the mean value inequality for the conjugate heat equation, at last, using above we derive the heat kernel bound of Gaussian type. On the other hand, assuming the boundedness of the Scalar curvature as before, we prove the backward pseudolocality property for the Ricci harmonic flow, combined with the muti-scale induction argument, we get the uniform control of the integral norm of the Riemannian curvature operator. This can be used to study the asymptotic behaviour and extension problem of the Ricci harmonic flow, and is closely related to the finite time blow up conjecture of the scalar curvature. 2. We consider Ricci harmonic soliton. On one hand, we study the decay estimate of its scalar curvature and compactness under the positive curvature condition. On the other hand, we study the compactness under various conditions, these will be important to study the asymptotic behaviour at infinity and classification.