Different curvature flows and their long time behaviour

本文主要研究欧几里得空间中不同抛物流的长时间特性，如Ricci流、Yamabe流和不同曲率的超曲面流。更准确地说，提议者想要理解可能的奇异极限度量的结构。由于古代解是以有限时间奇点的奇异模型出现的，因此建议研究不同流动情况下的奇异解的性质和分类。古代解的一个特例是梯度收缩孤子。它们的几何性质有很多需要了解的，特别是在完全高维情况下，这有助于它们的分类。关于奇点1，作者还建议研究最优条件，在此条件下可以保证Ricci流和平均曲率流的光滑解的存在。由于抛物型几何流的抛物性倾向于改善初始几何对象的性质，因此对不同抛物型几何流的研究非常感兴趣。例如，在初始度规的特定条件下里奇流趋向于永远存在并收敛于一个恒定截面曲率的度规这告诉我们很多关于流形拓扑的信息。这意味着人们有时可以利用抛物线几何流来解决其他数学领域的一些问题。古老的解是从负无穷开始的解。物理学家对理解里奇流的解很感兴趣。

The proposer is interested in a long time behaviour of different parabolic flows, such as the Ricci flow, the Yamabe flow and different curvature flows of hypersurfaces in the euclidean space. More precisely, the proposer would like to understand the structure of possible singular limiting metrics one gets. Since the ancient solutions occur as singularity models of finite time singularities, the proposer suggests to study the properties and the classification of those in the case of different flows. One special case of ancient solutions are the gradient shrinking solitons. There is much to be understood about their geometric properties especially in the complete higher dimensional cases which can help the classification of those. Related to the singularities I the proposer also suggests studying the optimal conditions under which one can guarantee the existence of a smooth solution to e.g. the Ricci flow and the mean curvature flow.The proposer is interested in studying different parabolic geometric flows since their parabolic properties tend to improve the properties of the initial geometric objects. For example, under certain conditions on the initial metric the Ricci flow tends to exist forever and converges to a metric of constant sectional curvature which tells us a lot about the topology of our manifold. That means one can sometimes use the parabolic geometric flows in order to resolve some issues in other mathematical fields. Ancient solutions are the solutions that come from all the way from negative infinity. The physicists are interested in understanding those solutions to the Ricci flow.