##### Department of Mathematics,

University of California San Diego

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### Math 258 - Differential Geometry

## Xiaolong Li

#### UCI

## Ancient Solutions to the Ricci Flow in Higher Dimensions

##### Abstract:

It is well-known that the Ricci flow will generally develop singularities if one flows an arbitrary initial metric. Ancient solutions arise as limits of suitable blow-ups as the time approaches the singular time and thus play a central role in understanding the formation of singularities. By the work of Hamilton, Perelman, Brendle, and many others, ancient solutions are now well-understood in two and three dimensions. In higher dimensions, only a few classification results were obtained and many examples were constructed. In this talk, we show that for any dimension $n \geq 4$, every noncompact rotationally symmetric ancient $kappa$-solution to the Ricci flow with bounded positive curvature operator must be the Bryant soliton, extending a recent result of Brendle to higher dimensions. This is joint work with Yongjia Zhang.

Host: Lei Ni

### February 6, 2019

### 1:00 PM

### AP&M 5829

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