On short time existence for the planar network flow

@article{Ilmanen2019OnST,
  title={On short time existence for the planar network flow},
  author={Tom Ilmanen and Andre' Neves and Felix Schulze},
  journal={Journal of Differential Geometry},
  year={2019}
}
We prove the existence of the flow by curvature of regular planar networks starting from an initial network which is non-regular. The proof relies on a monotonicity formula for expanding solutions and a local regularity result for the network flow in the spirit of B. White's local regularity theorem for mean curvature flow. We also show a pseudolocality theorem for mean curvature flow in any codimension, assuming only that the initial submanifold can be locally written as a graph with… 

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References

SHOWING 1-10 OF 25 REFERENCES

Self similar expanding solutions of the planar network flow

We prove the existence of self-similar expanding solutions of the curvature flow on planar networks where the initial configuration is any number of half-lines meeting at the origin. This generalizes

Motion by curvature of planar networks II

We prove that the curvature flow of an embedded planar network of three curves connected through a triple junction, with fixed endpoints on the boundary of a given strictly convex domain, exists

Motion by Curvature of Planar Networks

We prove that the curvature flow of an embedded planar network of three curves connected through a triple junction, with fixed endpoints on the boundary of a given strictly convex domain, exists

A general regularity theory for weak mean curvature flow

We give a new proof of Brakke’s partial regularity theorem up to $$C^{1,\varsigma }$$ for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic

Translating solutions to Lagrangian mean curvature flow

We prove some non-existence theorems for translating solutions to Lagrangian mean curvature flow. More precisely, we show that translating solutions with an $L^2$ bound on the mean curvature are

Curvature evolution of nonconvex lens-shaped domains

Abstract We study the curvature flow of planar nonconvex lens-shaped domains, considered as special symmetric networks with two triple junctions. We show that the evolving domain becomes convex in

Uniqueness and pseudolocality theorems of the mean curvature flow

Mean curvature flow evolves isometrically immersed base manifolds $M$ in the direction of their mean curvatures in an ambient manifold $\bar{M}$. If the base manifold $M$ is compact, the short time

Generic mean curvature flow I; generic singularities

It has long been conjectured that starting at a generic smooth closed embedded surface in R^3, the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the

Finite Time Singularities for Lagrangian Mean Curvature Flow

Given any embedded Lagrangian on a four dimensional compact Calabi-Yau, we find another Lagrangian in the same Hamiltonian isotopy class which develops a finite time singularity under mean curvature

Stratification of minimal surfaces, mean curvature flows, and harmonic maps.

In this paper we prove some abstract Stratification theorems for closed sets or more generally for upper semicontinuous functions. (Ciosed sets correspond to 0-1 valued upper semicontinuous