##### Department of Mathematics,

University of California San Diego

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### Math 258: Differential Geometry

## Jingze Zhu

#### MIT

## Spectral quantization for ancient asymptotically cylindrical flows

##### Abstract:

asymptotically cylindrical flows are ancient solutions to the mean curvature flow whose tangent flow at $-\infty$ are shrinking cylinders. In this talk, we study quantized behavior of asymptotically cylindrical flows. We show that the cylindrical profile function u of these flows have the asymptotics $u(y,\omega, \tau) = \frac{y^{T}Qy - 2tr Q}{|\tau|} + o(|\tau|^{-1})$ as $\tau\rightarrow -\infty$, where $Q$ is a constant symmetric $k\times k$ matrix whose eigenvalues are quantized to be either 0 or $-\frac{\sqrt{2(n-k)}}{4}$. Assuming non-collapsing, we can further draw two applications. In the zero rank case, we obtain the full classification. In the full rank case, we obtain the $SO(n-k+1)$ symmetry of the solution. This is joint work with Wenkui Du.

Host: Lei Ni

### January 19, 2023

### 11:00 AM

APM 7321

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