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# Ancient gradient flows of elliptic functionals and Morse index

## Christos Mantoulidis MIT

We study closed ancient solutions to gradient flows of elliptic functionals in Riemannian manifolds, focusing on mean curvature flow for the talk. In all dimensions and codimensions, we classify ancient mean curvature flows in $\mathbb{S}^n$ with low area: they are steady or canonically shrinking equators. In the mean curvature flow case in $\mathbb{S}^3$, we classify ancient flows with more relaxed area bounds: they are steady or canonically shrinking equators or Clifford tori. In the embedded curve shortening case in $\mathbb{S}^2$, we completely classify ancient flows of bounded length: they are steady or canonically shrinking equators. (Joint with Kyeongsu Choi.)

Seminario 1ª planta, IEMath-GR

# Christos Mantoulidis

## MIT

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