My question is about the compactness argument to find a limiting self-similar solution to the mean curvature flow near a singularity.

Consider a mean curvature flow $(M_t)_{t \in I}$ of $n$-dimensional immersed manifolds in $\mathbb{R}^{n+1}$ developing a type I singulartiy at the oringin at time $T$. We can rescale these surfaces as follows $$ \tilde{F}(p,s) = \frac{1}{\sqrt{2(T-t)}} F(p,t), \: \: \: s(t) = -\frac{1}{2} \text{ln}(T-t). $$ It is well known that all the covariant derivatives of the second fundamental form of $\tilde{F}$ are bounded for $s \in [-\frac12 \text{ln}(T), \infty)$. Then I would like to ask why Huisken, in his famous paper about singularities, used an unknown (at least to me) compactness theorem to show that $\tilde{F}$ tends, locally, to a smooth immersion for some subsequence $(s_j)$. Since to me, it doesn't seem to hard to show the boundedness of the Christoffel symbols and of the metric for $\tilde{F}$ for all points $p$ in a compact set where $\tilde{F}(p,s)$ remains bounded for $s \rightarrow \infty$. Then one can use the well-known Ascoli-Arzela theorem together with a diagonal subsequence argument to find such a smooth limiting surface.

Edit: While thinking about this I found that proving the boundedness of the Christoffel symbols could in fact be not so easy. This is since the symbols of the original imersion evolves as $$ \frac{\partial \Gamma_{ij}^k}{\partial t} = A \ast \nabla A, $$ where the $\ast$ denotes a metric contraction of the two tensors. Also we know that $A$ blows up somewhere near a singularity and together with the fact that the christoffel symbols of the rescaled immersion are the same ones as the original immersion we may indeed get stuck.

But still I'de like to have some explanation or confirmation why Huisken didn't use the Ascoli-Arzela compactness theorem.

  • $\begingroup$ Related. $\endgroup$ – user99914 Nov 10 '17 at 4:30

This is one part in mean curvature flow where no one cares to write down the proof but everyone is using it.

I don't think a complete proof of the compactness theorem is available when Huisken uses it in 1990. However, a similar one (that he did cite) is given by J. Langer, where the case of surfaces in $\mathbb R^3$ is considered. I have a feeling that everyone believes that it is true and have the same idea of proof in mind (using Ascoli-Arzela theorem).

Later the compactness theorem is generalized to immersions $f : M^n \to \mathbb R^{n+1}$. The general case (any dimension, codimension in euclidean space) is proved recently by P. Breuning. The idea is that if one write locally the immersion as a graph $(x, f(x))$, then the second derivative of $f$ can be bounded by the second fundamental form (see Lemma 2.2 here).

In general it is not true that the second derivative of an immersion $F : \Omega \to \mathbb R^K$ is bounded by second fundamental form - after all, the second fundamental form does not depend on $F$, but only on the geometry of $F(\Omega)$. So the above lemma is only true in the graphical case. And this mean when we say that $\tilde F (\cdot, s_j)$ converges to some smooth immersion, it is understood that we have to compose each $\tilde F(\cdot, s_j)$ with some diffeomorphism $\phi_j : \Omega_j \to \Omega_j$, where $\Omega_j$ is some domain inside $M$.

In all of the above works, they are not using Ascoli-Arzela theorem since they have only $L^p$-bound instead of $C^0$ bound. But if one have $C^0$ bound, then I believe one can copy their proof and use Ascoli-Arzela theorem instead.

  • $\begingroup$ Thanks for the clarification! I've read the proof given by J. Langer, I think some constructions in the proof are a little bit easier in our case since we have uniform bounds on all the covariant derivatives of the second fundamental form. Furthermore is this theorem needed since, as in the question, the Christoffel symbols of the original immersion may become unbounded and hence we can't control the derivatives of the original immersion? $\endgroup$ – abcdef Nov 4 '17 at 9:29
  • $\begingroup$ What you need is really something like Lemma 2.2, where one relate curvature bounds to bound of a graphical function $f$. @abcdef . I don't think one can bound the Christoffel symbol directly. $\endgroup$ – user99914 Nov 5 '17 at 3:35

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