On short time existence for the planar network flow

TL;DR

This paper establishes short-time existence results for the curvature flow of planar networks starting from non-regular initial configurations, using monotonicity formulas and local regularity techniques.

Contribution

It introduces a method to prove short-time existence for network flow from non-regular initial networks, extending previous results.

Findings

Existence of flow by curvature for non-regular initial networks

Development of a monotonicity formula for expanding solutions

A pseudolocality theorem for mean curvature flow in arbitrary codimension

Abstract

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 sufficiently small Lipschitz constant.