Ricci Solitons Days in Pisa 2011

Initial Time Singularities in Mean Curvature Flow

speaker: Tom Ilmanen (ETH)

abstract: Let M₀ be a closed subset of Rⁿ⁺¹ that is a smooth hypersurface except for a finite number of isolated singular points. Suppose that M₀ is asymptotic to a regular cone near each singular point.

Can we flow M₀ by mean curvature?

Theorem (n<7): there exists a smooth mean curvature evolution starting at M₀ and defined for a short time \( 0< t < \varepsilon \).

Such an initial M₀ might arise as the limit of a smooth mean curvature evolution defined earlier than t=0. Thus, the result allows us to flow through singularities in some cases.

We use a monotonicity formula that complements the monotonicity formula of Huisken. The method applies to other geometric heat flows as well.

timetable:
Tue 5 Apr, 14:30 - 15:30, Aula Dini
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