Evolution of spoon-shaped networks

Abstract

We consider a regular embedded network composed by two curves, one of them closed, in a convex domain Ω. The two curves meet only in one point, forming angle of 120 degrees. The non-closed curve has a fixed end point on ∂Ω. We study the evolution by curvature of this network. We show that the maximal existence time depends only on the area enclosed in the initial loop, if the length of the non-closed curve stays bounded from below during the evolution. Moreover, the closed curve shrinks to a point and the network is asymptotically approaching, after dilations and extraction of a subsequence, a Brakke spoon.Mathematics Subject classification (2010). 53C44 (primary); 53A04, 35K55 (secondary).with loops. We call Brakke spoon a spoon-shaped network composed by a half-line which meets a closed curve shrinking in a self-similar way during the evolution. The main result is the following: Theorem 1.1. Given a spoon-shaped network Γ t evolving by curvature in a strictly convex and open set Ω ∈ R 2 , there exists a finite time T such that either the inferior limit of the length of the non-closed curve goes to zero as t → T , or there exists a point x 0 ∈ Ω such that for a subsequence of rescaled time t j the associate rescaled network tends to a Brakke spoon as j → ∞.This result can be seen as an analogue of Grayson's theorem for a closed curve, which says that an embedded evolving curve in R 2 becomes eventually convex without developing singularities and then shrinks to a point (see [6,7]). A new proof of this theorem is given in [14]. In Proposition 5.4, we obtain, up to subsequence, as possible limits, sets that satisfy the equation k + x|ν = 0 for all x, where k is the curvature and ν is the unit normal vector. Satisfying the previous equation, the sets shrink homothetically. The existence and uniqueness of self-similar shrinking solution of a problem similar to ours is proved in [4]: Chen and Guo consider an equivalent system that describes the motion by curvature, but they focus on the evolution of a curve symmetric about the x−axis, and consider the part of the curve in the upper half-plane, which forms fixed angles with the axis. In Proposition 5.4, to gain existence of a unique limit set, without self-intersections, with multiplicity one, with curvature not constantly zero, we will apply [4, Theorem 3], as our asymptotic solution is one of the Abresch-Langer curves, symmetric and convex.

Networks self-similarly moving by curvature with two triple junctions

“…Recently, the problem of the evolution by curvature of a network of curves in the plane got the interest of several authors [3,7,11,12,[15][16][17][18][19]. It is well known, after the work of Huisken [8] in the smooth case of the hypersurfaces in the Euclidean space and of Ilmanen [9,10] in the more general weak settings of varifolds, that a suitable sequence of rescalings of the subsets of R n which are evolving by mean curvature, approaching a singular time of the flow, converges to a so called "blow-up limit" set which, letting it flow again by mean curvature, simply moves by homothety, precisely, it shrinks down self-similarly toward the origin of the Euclidean space.…”

Motion by curvature of networks with two triple junctions

“…The simplest case of a network with a loop (a region bounded by one or more curves) is treated in [21]: a network composed by two curves, one of them closed, meeting only at one point. In this case, if the length of the non-closed curve is bounded away from zero during the evolution, the closed curve shrinks to a point after a finite time which depends only on the area initially enclosed in the loop.…”

Stability of regular shrinkers in the network flow

“…In [11], Epstein and Weinstein use a perturbation in the space of curvature to show that only the circle with multiplicity 1 is the only stable compact self-shrinker in R 2 . The Brakke spoon is shown to be the blow-up limit for all spoon-shaped networks in the work of Pluda [20]. This implies stability since any perturbation of the Brakke spoon is topologically spoonshaped.…”