Complexity of parabolic systems

Abstract

We first bound the codimension of an ancient mean curvature flow by the entropy. As a consequence, all blowups lie in a Euclidean subspace whose dimension is bounded by the entropy and dimension of the evolving submanifolds. This drastically reduces the complexity of the system. We use this in a major application of our new methods to give the first general bounds on generic singularities of surfaces in arbitrary codimension.We also show sharp bounds for codimension in arguably some of the most important situations of general ancient flows. Namely, we prove that in any dimension and codimension any ancient flow that is cylindrical at −∞ must be a flow of hypersurfaces in a Euclidean subspace. This extends well-known classification results to higher codimension.The bound on the codimension in terms of the entropy is a special case of sharp bounds for spectral counting functions for shrinkers and, more generally, ancient flows. Shrinkers are solutions that evolve by scaling and are the singularity models for the flow.We show rigidity of cylinders as shrinkers in all dimension and all codimension in a very strong sense: Any shrinker, even in a large dimensional space, that is sufficiently close to a cylinder on a large enough, but compact, set is itself a cylinder. This is an important tool in the theory and is key for regularity; cf. [CM11].

Liouville properties

“…This is a subject that has been notoriously difficult and where much less is known than for hypersurfaces. The idea of [CM10] is to use ideas described in the earlier sections to show that blowups of higher codimension MCF have codimension that typically is much smaller than in the original flow. In many important instances we can show that blowups are evolving hypersurfaces in an Euclidean subspace even when the original flow is very far from being hypersurfaces.…”

“…The spectral counting function N (µ) is the number of eigenvalues µ i ≤ µ counted with multiplicity. In [CM10] we bound the spectral counting function for any shrinker in terms of n, λ(Σ), and µ. As a special case we get:…”

“…Theorem 7.9. (Bounding codimension by entropy for shrinkers, [CM10]). If Σ n ⊂ R N is a shrinker, then Σ ⊂ an Euclidean subspace of dimension ≤ C n λ(Σ).…”

“…Theorem 7.11. (Sharp codimension bound, [CM10]). If M n t ⊂ R N is an ancient MCF and one tangent flow at −∞ is a cylinder, then M t is a flow of hypersurfaces in a Euclidean subspace.…”

Nonplanar ancient curve shortening flows in $\mathbb{R}^3$ from grim reapers

“…There are many known examples of ancient CSF/MCF in R n , many of which are nonplanar, see for instance [1,9]. The quality of nonplanarity is interesting for instance because there are a number of recent results for the mean curvature in higher codimension where one can constrict ancient solutions to some proper affine subspace of R n under some conditions, see for example [6,8,10]. In particular, a nonplanar curve shortening flow in R 3 takes up as much "room" as possible.…”