Different curvature flows and their long time behaviour

提出者感兴趣的是不同的抛物流，如Ricci流，Yamabe流和不同曲率流的超曲面在欧氏空间中的长时间行为。更确切地说，提议者希望了解可能的奇异极限度量的结构。由于古代解是作为有限时间奇点的奇点模型出现的，建议研究不同流动情况下的解的性质和分类。古解的一个特例是梯度收缩孤子。关于它们的几何性质还有很多东西需要理解，特别是在完全高维的情况下，这有助于对它们进行分类。与奇点I相关，提议者还建议研究最优条件，在该条件下，可以保证例如Ricci流和平均曲率流的光滑解的存在性。提议者对研究不同的抛物几何流感兴趣，因为它们的抛物性质倾向于改善初始几何对象的性质。例如，在初始度量的某些条件下，里奇流趋于永远存在，并收敛到一个截面曲率为常数的度量，这告诉了我们很多关于流形拓扑的信息。这意味着人们有时可以使用抛物几何流来解决其他数学领域的一些问题。古代的解是从负无穷远处来的解。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.