Advances in Mathematics ( IF 1.7 ) Pub Date : 2021-01-26 , DOI: 10.1016/j.aim.2021.107621 Francesco Di Giovanni
We study the Ricci flow on , with , starting at some complete bounded curvature rotationally symmetric metric . We first focus on the case where does not contain minimal hyperspheres; we prove that if is asymptotic to a cylinder, then the solution develops a Type-II singularity and converges to the Bryant soliton after scaling, while if the curvature of decays at infinity, then the solution is immortal. As a corollary, we prove a conjecture by Chow and Tian about Perelman's standard solutions. We then consider a class of asymptotically flat initial data containing a neck and we prove that if the neck is sufficiently pinched, in a precise way, the Ricci flow encounters a Type-I singularity.
中文翻译:
旋转对称Ricci流
我们研究Ricci流 ,带有 ,从某个完全有界曲率旋转对称度量开始 。我们首先关注的情况是不包含最少的超球面;我们证明 对圆柱渐近,则解产生II型奇点并在缩放后收敛到科比孤子,而如果 衰减到无穷远,那么解法是不朽的。作为推论,我们证明了Chow和Tian对Perelman的标准解决方案的猜想。然后,我们考虑一类渐近平坦的初始数据 包含一个脖子,我们证明如果脖子被精确地充分捏住,则Ricci流会遇到I型奇点。