Motion by curvature of networks with two triple junctions

Abstract

Abstract:We consider the evolution by curvature of a general embedded network with two triple junctions. We classify the possible singularities and we discuss the long time existence of the evolution.MSC: 53C44 (primary); 53A04, 35K55 (secondary)

Networks self-similarly moving by curvature with two triple junctions

“…We conclude this discussion mentioning that the main motivation for this problem is given by the fact that for an evolving network with at most two triple junctions, the so called multiplicity-one conjecture holds (see [14]), saying that any limit shrinker of a sequence of rescalings of the network at different times is again a "genuine" embedded network without "double" or "multiple" curves (curves that in such convergence go to coincide in the limit). This is a key point in the singularity analysis (actually, in general, for mean curvature flow), together with the classification of these limit shrinkers, which is complete after our result Theorem 1.1, for such "low complexity" networks, thus leading to a detailed description of their motion in [13].…”

“…It is then known, by the work of Abresch-Langer [1] and independently of Epstein-Weinstein [6], that the only complete, embedded, self-similarly shrinking curves in R 2 without end-points, are the lines through the origin and the unit circle (they actually classify all the closed, embedded or not, self-similarly shrinking curves in the plane). The same equation k + γ ⊥ = 0 (that is, k + γ | ν = 0) must be satisfied by every curve of a network in the plane which self-similarly shrinks to the origin moving by curvature (see [14,15], for instance). Moreover, for "energetic" reasons, it is natural to consider networks with only triple junctions and such that the three concurring curves (which are C ∞ ) form three angles of 120 degrees between each other -"Herring" condition -such networks are called regular.…”

On the Classification of Networks Self–Similarly Moving by Curvature

“…This is a key point in the singularity analysis (actually, in general, for mean curvature ow), together with the classi cation of these limit shrinkers, which is complete after our result Theorem 1.1, for such "low complexity" networks, thus leading to a detailed description of their motion in [15].…”

“…We conclude this discussion mentioning that the main motivation for this problem is given by the fact that for an evolving network with at most two triple junctions, the so called multiplicity-one conjecture holds (see [16]), saying that any limit shrinker of a sequence of rescalings of the network at di erent times is again a "genuine" embedded network without "double" or "multiple" curves (curves that in such a convergence go to coincide in the limit). This is a key point in the singularity analysis (actually, in general, for mean curvature ow), together with the classi cation of these limit shrinkers, which is complete after our result Theorem 1.1, for such "low complexity" networks, thus leading to a detailed description of their motion in [15].…”

“…of a network in the plane which self-similarly shrinks to the origin moving by curvature (see [16,17], for instance). Moreover, for "energetic" reasons, it is natural to consider networks with only triple junctions and such that the three concurring curves (which are C ∞ ) form three angles of degrees between each other -"Herring" condition -such networks are called regular.…”