In Huisken, Gerhard, Asymptotic behavior for singularities of the mean curvature flow, there is a conjecture that the blow up rate of singularities of arbitrary embedded surfaces is same with convex surfaces. (1) $$ \text{blow up rate: }~~~ \frac{1}{2(T-t)}\le \max_{M_t}|A|^2\le \frac{C_0}{2(T-t)} ~~~~~~~ $$

And Huisken explains it is not right for all immersed surfaces. Because the curvature of an immersed curve developing a cusp will blow up at a faster rate. (2)

First, about the explain (2), there is not a analytic explain , how to know it has a faster rate?

Second, I treat the embedding as immersion adding homeomorphism. Why the homeomorphism can eliminate the situation in (2) ?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.