Networks self-similarly moving by curvature with two triple junctions

Baldi, Pietro; Haus, Emanuele; Mantegazza, Carlo

doi:10.4171/rlm/765

Cited by 7 publications

(46 citation statements)

References 13 publications

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“…For each j, since on an Abresch and Langer curve, ke − |x| 2 4 is a constant, we have ∇(ke − |x| 2 4 ) j i = 0 for each j. Therefore, 4 ), T j i = 0 and it is independent of j. For the function V, N , we have…”

Section: The Eigenfunction Problem From the 2nd Variation Formulamentioning

confidence: 98%

“…(3) f is constant on each ray which goes to infinity. From lemma 8.1, the contribution of any curve γ is given by γ −1 4 dσ. There are two cases, γ is a ray or not.…”

Section: The Eigenfunction Problem From the 2nd Variation Formulamentioning

confidence: 99%

“…The classification of regular shrinker with one enclosed region is done by Chen and Guo [9]. Baldi, Haus, and Mantegazza [4,5] exclude the Θ-shaped network. Following their work, the author and Lue [10] show that there is only one regular shrinker with exactly two enclosed regions.…”

Section: Introductionmentioning

confidence: 99%

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Stability of regular shrinkers in the network flow

Chang

2021

Preprint

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Section: The Eigenfunction Problem From the 2nd Variation Formulamentioning

confidence: 98%

“…(3) f is constant on each ray which goes to infinity. From lemma 8.1, the contribution of any curve γ is given by γ −1 4 dσ. There are two cases, γ is a ray or not.…”

Section: The Eigenfunction Problem From the 2nd Variation Formulamentioning

confidence: 99%

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Stability of regular shrinkers in the network flow

Chang

2021

Preprint

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“…Also the classification of shrinkers with two triple junctions is complete. It is not difficult to show [6,7] that there are only two possible topological shapes for a complete embedded, regular shrinker: one is the "lens/fish" shape and the other is the shape of the Greek "Theta" letter (or "double cell"). It is well known that there exist unique (up to a rotation) lens-shaped or fish-shaped, embedded, regular shrinkers which are symmetric with respect to a line through the origin of R 2 [13,38] (Figure 8).…”

Section: Self-similar Solutionsmentioning

confidence: 99%

Lectures on Curvature Flow of Networks

Mantegazza

Novaga

Pluda

2019

Contemporary Research in Elliptic PDEs and Related Topics

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We present a collection of results on the evolution by curvature of networks of planar curves. We discuss in particular the existence of a solution and the analysis of singularities.The aim of this work is to provide an overview on the motion by curvature of a network of curves in the plane. This evolution problem attracted the attention of several researchers in recent years, see for instance [9-11, 13, 20, 23, 24, 29, 31, 33, 36, 38, 44]. We refer to the extended survey [32] for a motivation and a detailed analysis of this problem.This geometric flow can be regarded as the L 2 -gradient flow of the length functional, which is the sum of the lengths of all the curves of the network (see [10]). From the energetic point of view it is then natural to expect that configurations with multi-points of order greater than three or 3-points with angles different from 120 degrees, being unstable for the length functional, should be present only at a discrete set of times, during the flow. Therefore, we shall restrict our analysis to networks whose junctions are composed by exactly three curves, meeting at 120 degrees. This is the so-called Herring condition, and we call regular the networks satisfying this condition at each junction.The existence problem for the curvature flow of a regular network with only one triple junctions was first considered by L. Bronsard and F. Reitich in [11], where they proved the local existence of the flow, and by D. Kinderlehrer and C. Liu in [24], who showed the global existence and convergence of a smooth solution if the initial network is sufficiently close to a minimal configuration (Steiner tree).We point out that the class of regular networks is not preserved by the flow, since two (or more) triple junctions might collide during the evolution, creating a multiple junction composed by more than three curves. It is then natural to ask what is the subsequent evolution of the network. A possibility is restarting the evolution at the collision time with a different set of curves, describing a non-regular network, with multi-points of order higher than three. A suitable short time existence result has been worked out by T. Ilmanen, A. Neves and F. Schulze in [23], where it is shown that there exists a flow of networks which becomes immediately regular for positive times.1. the interior of every curve is injective, a curve can self-intersect only at its end-points; 2. two different curves can intersect each other only at their end-points;3. a curve is allowed to meet ∂Ω only at its end-points; 4. if an end-point of a curve coincide with P ∈ ∂Ω, then no other end-point of any curve can coincide with P .The curves of a network can meet at multi-points in Ω, labeled by O 1 , O 2 , . . . , O m . We call end-points of the network, the vertices (of order one) P 1 , P 2 , . . . , P l ∈ ∂Ω.Condition 4 keeps things simpler implying that multi-points can be only inside Ω, not on the boundary.We say that a network is of class C k with k ∈ {1, 2, . . .} if all its curves are of class C k .

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“…From the work of Mantegazza, Novaga, and Pluda [14], for an evolving network with at most two triple junctions, the multiplicity-one conjecture holds. P. Baldi, E. Haus, and Mantegazza [4,5] exclude the Θ-shaped network. Together with the work by Chen and Date: February 13, 2019.…”

Section: Introductionmentioning

confidence: 99%