Type-0 singularities in the network flow – Evolution of trees
-
Carlo Mantegazza
,
Matteo Novaga
and
Alessandra Pluda
Abstract
The motion by curvature of networks is the generalization to finite union of curves of the curve shortening flow. This evolution has several peculiar features, mainly due to the presence of junctions where the curves meet. In this paper we show that whenever the length of one single curve vanishes and two triple junctions coalesce, then the curvature of the evolving networks remains bounded. This topological singularity is exclusive of the network flow and it can be referred as a “Type-0” singularity, in contrast to the well known “Type-I” and “Type-II” ones of the usual mean curvature flow of smooth curves or hypersurfaces, characterized by the different rates of blow-up of the curvature. As a consequence, we are able to give a complete description of the evolution of tree–like networks till the first singular time, under the assumption that all the “tangents flows” have unit multiplicity. If the lifespan of such solutions is finite, then the curvature of the network remains bounded and we can apply the results from [T. Ilmanen, A. Neves and F. Schulze, On short time existence for the planar network flow, J. Differential Geom. 111 2019, 1, 39–89] and [J. Lira, R. Mazzeo, A. Pluda and M. Saez, Short–time existence for the network flow, preprint 2021] to “restart” the flow after the singularity.
1 Introduction
One of the most studied geometric evolution equations is the mean curvature flow: a family of time-dependent n-dimensional hypersurfaces in
The one-dimensional version of such multiphase mean curvature flow is usually called motion by curvature of networks or network flow, that is, a network of curves moving in the plane decreasing the total length in the “most efficient” way and hopefully converging to a configuration of minimal length (or at least a “critical” configuration for the “total length functional” on the networks).
There are basically two approaches to this problem, related to considering weak or strong (smooth) solutions, even if they have several points of contact. An advantage of the weak formulations is that one gets easily global existence results, but then showing uniqueness and regularity becomes quite hard. On the other side, well posedness of strong solutions is easier, but it is then complicated to define a flow passing past the singularities.
Different notions of weak solutions have been investigated by several authors, see
for instance [3, 5, 9, 12, 14].
A very relevant one is the weak definition of the flow given in the seminal work by Brakke [5], that is, a family of n-dimensional varifolds
The long term goal of our project is a complete description of the network flow defined in the framework of classical PDEs, that is, we are interested in smooth solutions far from singular times. We consider the evolution of a finite union of smooth curves meeting at multiple junctions, since the network flow is the gradient flow of the length, it is expected (by “energetic” reasons) that for almost all times the evolving networks present only triple junctions where the unit tangent vectors of the three concurring curves sum up to zero. Hence, it is natural to first to restrict the analysis to the class of “regular” networks, where curves meet only at triple junctions forming angles of 120 degrees. Then, the motion can be written as a system of quasilinear second order parabolic equations with nonlinear boundary conditions (see (2.1)).
The first well-posedness result for this problem has been established in the 1990s
by Bronsard and Reitich [6] whenever the initial network (all its curves) is of class
Since by these results the flow starts, we can then study the long time behavior of the solutions.
By [21, Theorem 3.18], if the maximal time of existence of a solution is finite, then
either the curvature blows up or the length of at least one curve goes to zero.
For instance, the area of a region bounded by less then six curve decrease linearly in time, then when the region vanishes, it can be shown that the curvature blows up. It is also easy to find an example where only he length of a single curve goes to zero and either
one triple junction collapses onto one boundary point or two triple junctions coalesce.
However, there are no examples in which the lengths of all the curves
of the network remain strictly positive and the curvature goes to
Multiplicity-One Conjecture (M1).
Every limit of parabolic rescalings of the flow around a fixed point in
We are indeed able to prove that it if no length goes to zero and this conjecture is true, then the curvature remains bounded during the flow. In other words, there are no singularities until at least one curve vanishes.
As we said before, if a region disappears at the maximal time of existence, then the curvature cannot remain bounded. One may guess that if the initial network does not contain any loop, hence there are no regions that can collapse, this type of singularities are excluded. We actually have a positive answer.
Theorem 1.1.
Let
We briefly outline the proof of this result.
We consider suitable rescaled flows around points of
It remains to show the same conclusion when the limit is a standard cross, and we will show this in Theorem 5.4. First, in Lemma 4.2 and 4.3 we prove that for any tree, if we assume a uniform control on the motion of its end-points, the
Even if the curvature is then always controlled, in Section 6 we show that singularities might still occur during the evolution, due to topological changes when the length of a curve goes to zero. Indeed, we explicitly construct an example of a network composed of five curves where two triple junctions coalesce in finite time with the vanishing of one curve. Because of their peculiar nature, we refer to these singularities with bounded curvature as Type-0 singularities (suggestion by Tom Ilmanen). We stress the fact that they are exclusive of the network flow, as they cannot appear in the motion by curvature of a single curve.
It is worth mentioning that, if a network contains at most two triple junctions, we can prove that every limit of parabolic rescalings is a flow of embedded networks with multiplicity one [19], but unfortunately the Multiplicity-One Conjecture is still an open problem in full generality. Assuming it, we have a quite complete, “nice” description of the flow of tree-like networks till the first singular time.
Theorem 1.2.
Let
We mention that then one can apply the results in [11, 16] and “restart” the flow with a smooth evolving regular network, hence the main question that remains open in order to obtain a global existence result, is the possible accumulation in time of the singularities, that we will not address in this paper.
2 Preliminaries
2.1 Notation
Consider a curve of class
Assuming that the
curve is of class
where κ is the scalar curvature The arclength parameter of a curve γ is given by
We remind here that
in the whole paper we will use the word “curve” both for the parametrization and for the
set image of the parametrization in
Definition 2.1.
A network
We distinguish between interior and exterior vertices of the network: at the firsts, more then one curve concur, and the latter are the termini of the network.
We say that a network is a tree if it does not contain loops.
We assume that each curve of a network admits a regular parametrization
All along the paper we denote by
We are going to give a name to the following special sub-class of networks because of the key role they are going to play in the paper:
Definition 2.2.
A network is regular if it possesses only triple junctions, where the unit tangent vectors form angles of 120 degrees.
Consider a regular network
composed of N curves
2.2 Motion by curvature of networks
We are interested in the motion by curvature of networks and to deal with this problem we use a direct PDE approach. The network flow can be understood as the gradient flow of the length
Formally, we derive the motion equations computing the first variation of L. Each curve moves with normal velocity equal to its curvature
or, equivalently
Let Ω be a convex, open, bounded and regular set in
Definition 2.3.
Let Ω be a bounded, convex, open and regular subset of
that satisfy
the following system of conditions for every
where we assumed conventionally (possibly relabeling the family of
curves and inverting their parametrization) that the end-point
Note that in the third equation we write
We underline the fact that the motion equation is equivalent to
where ζ is a nonzero tangential velocity, at the junctions geometrically determined by the curvatures of the concurring curves. In the whole paper we will label the tangential velocity by ζ. When
we obtain the so-called special flow
2.3 Short time existence and behavior at the maximal time of existence
Combining results from [6, 10, 20], one gets:
Theorem 2.4.
Let
Proof.
The main result contained in [6] is existence and uniqueness
of the flow in a small time interval
A short time existence result can be shown, with some appropriate weak definitions, also to initial
An example of non-uniqueness of the flow.
Once established the existence of a solution in a maximal time interval
Proposition 2.5.
Let
or at least one of the following properties holds:
the inferior limit of the length of at least one curve of the network
the superior limit of the integral of the squared curvature is
Moreover, if the lengths of all the curves of the network are uniformly
bounded from below during the flow,
then the superior limit is a limit, and for every
2.4 Monotonicity formula and Multiplicity-One Conjecture
The problem we shall tackle now is analyze possible singularities arising at the maximal time of existence. To this end, we introduce a rescaling procedure and a monotonicity formula, adapted to our situation and originally due to Huisken, that enable us to better describe the singularities.
To determine and predict how singularities form, homothetically shrinking solutions play an important role.
Definition 2.6.
A time-dependent family of networks
with
Plugging our ansatz (2.2) into the motion equation
Definition 2.7.
The last relation is called the shrinker equation. A (static) network whose curves satisfy the equation
is called shrinker.
Let us introduce a parabolic rescaling procedure and the notion of reachable points (see [20, Section 7, Section 10] for more details).
We can describe the solution to the motion by curvature
as a map from a reference network
We define the set of reachable points of the flow as
Such a set is easily seen to be closed and contained in
Definition 2.8.
For a fixed
where
After this rescaling we still have that
Let
Definition 2.9.
For every
and, provided
which exists, is finite and nonnegative for every
Given a sequence
where
and
This result is nontrivial, also because of the presence of the boundary terms, we refer to [20, Section 7] for further details.
The monotonicity formula is the key ingredient to obtain the following:
Proposition 2.10.
Let
Proof.
See [20, Proposition 8.16]. ∎
The flows obtained as a limit of parabolically rescaled curvature flows (see Definition 2.8) go under the name of tangent flow. Now we would like to classify the possible self-similarly shrinking network flows arising as limit of parabolically rescaled curvature flows. We state now a conjecture about their nature.
Multiplicity-One Conjecture (M1).
Every limit of parabolic rescalings at a point
Theorem 2.11.
Let
either
or
Then the Multiplicity-One Conjecture holds true.
3 Analysis of tangent flows
From now on we shall always assume that the Multiplicity-One Conjecture holds true.
Definition 3.1.
A standard triod is a network composed of three straight halflines that meet at the origin forming angles of 120 degrees. A standard cross is the union of two lines intersecting at the origin forming angles of 120 and 60 degrees. (See Figure 2).
A standard triod and a standard cross.
Lemma 3.2.
Let
a line,
a standard triod,
a standard cross.
Proof.
First of all we notice that the length of some curves can go to zero in the limit. The core of the limit network is the union of these vanishing curves.
Suppose that the length of
at least one curve of
We can and will suppose (up to reparametrization)
that for every
Indeed, given
We then obtain the desired result by passing to the limit.
Hence the vanishing curves of
We describe now the structure of the core.
Let
If the unit tangent vector turns of an angle of 60 degrees in the same direction for two consecutive times,
Hence when we run across such longest path and we pass from one edge to another, if the unit tangent vector turns of an angle of 60 degrees, then it must turn back. This means that at the initial/final point of such path, either the two unit tangent vectors are the same (when the number of edges is odd) or they differ of 60 degrees (when the number of edges is even). By a simple check, we can then see that, in the first case the four curves with positive length exiting from such initial/final points of the path, have four different exterior unit tangent vectors at P (opposite in pairs), in the second case, they have three exterior unit tangent vectors at P which are non-proportional each other (see Figure 4).
Examples of the edges at the initial/final points of the longest simple path in
Consider the curves with positive length.
Each curve of the limit satisfies the shrinkers equation (2.3).
They have been classified by Abresch and Langer [1].
There are basically two possibilities.
A branch of the network is non-compact and it is
either a straight line through the origin or a straight halfline
with end-point the origin.
Otherwise, it is contained in a
compact subset of
Our scope at this point is to exclude the presence of compact branches in
Now we assume that
The possible local structure of the graphs G, with relative networks
Then we conclude that the only possible network with a core is when
Lemma 3.3.
Let
a static flow of a straight line through the origin,
a static flow of a standard triod centered at the origin,
a static flow of a standard cross centered at the origin.
If
a static flow of a halfline from the origin,
a static flow of two concurring halflines forming an angle of 120 degrees at the origin.
Proof.
By hypothesis we consider a sequence of parabolically rescaled curvature flows
There is a one-to-one correspondence between shrinkers and self-similar shrinking flow:
if
with
a straight line,
a standard triod,
a standard cross.
If instead
a halfline,
two concurring halflines forming an angle of 120 degrees.
We thus have listed all possible cases. ∎
If we add an extra assumption of the lengths of the curves of the evolving network, two of the previous cases are excluded. To be precise we have:
Corollary 3.4.
Let
Let
a static flow of a straight line through the origin (in this case
a static flow of a standard triod centered at the origin (in this case
If the rescaling point
a static flow of a halfline from the origin(in this case
If the rescaling point does not belongs to
Proof.
The possible limit flow is among the ones listed in the previous lemma. Moreover, the bound from below on the lengths exclude the possibility that the shrinker has a core. Clearly the only remaining options are the line, the standard triod and the halfline. ∎
3.1 Local regularity
Definition 3.5.
We say that a network
for every
Theorem 3.6.
Let
for all
then there exists
for all
Proof.
See [11, Theorem 1.3]. ∎
The following corollary is then an extension of White’s result [23, Theorem 3.5] to the curvature flow
Corollary 3.7.
Let
either at
or at
Then the curvature is uniformly bounded along the flow in a neighborhood of
Proof.
It is well known that in these hypotheses the family of networks
for some small
thus, again by the monotonicity (possibly choosing smaller
for any
for every
4 Estimates on geometric quantities
We list here several relations fulfilled by geometric quantities at the triple junctions of the solutions of smooth flow of regular networks. However before stating the lemma we list the evolution equation of several quantities used in the sequel.
If γ is a curve moving by
To show it, we let
Then, thanks to the commutation rule for an evolving curve, we can compute
where
Lemma 4.1.
Let
with the convention that the superscripts appearing in sums are to be considered “modulus 3”.
Proof.
Let
for every
that lead to
which implies
and so we obtained both (4.1) and (4.2).
Now we differentiate in time the angle condition
Thus
Now we show equation (4.5). Keeping in mind (4.1) and (4.2), we obtain
and
and so equality (4.5) is proved. ∎
Lemma 4.2.
Assume that
Proof.
Let us start from the simple case of a network
A tree-like network with five curves.
Fixed a time
hence, for every
Assume now instead that
Moreover, by (4.2) we have
Arguing as before, we get
In conclusion, we get the uniform in time inequality for
In the general case, since
Lemma 4.3.
Let
for every
Here the constant C depends only on the number
Proof.
By direct computations and relation (4.4) we have
By estimate (4.6) and the Young inequality, we then obtain
for every small
where we chose
Calling
hence, after integration, we get
and, choosing
We prove now that if we consider a tree-like network composed of five curves
with two triple junctions and four end-points on
Lemma 4.4.
Let
Proof.
We first estimate
Fixed a time
hence, in this case,
for every
Assume now instead that
hence,
by Lemma 4.3. Then
and it follows
since, by the previous argument, we have
for all
We now pass to estimate
Now we use (4.5) to lower the differentiation order of
the term
and from the fact that
we get
at the triple junctions
Substituting this equality into estimate (4.10), we obtain
Using the previous estimate on
for any small value
We deal now with the boundary term
hence,
Then we can write
At the triple junction
By hypothesis the length of the curves
via Gagliardo–Nirenberg interpolation inequalities. We get
hence,
Thus, finally,
Coming back to computation (4.11), we have
where we chose
By Gronwall’s Lemma, it follows that
is uniformly bounded, for
as
5 Behavior at singular times
The aim of this section is to improve Proposition 2.5.
We show that the length of at least one curve must go to zero.
Moreover, if we suppose that
5.1 Vanishing of curves
Proposition 5.1.
Let
Proof.
Assume that
We have just shown that if as
It remains to understand if as
5.2 The curvature remains bounded
We state here two lemma on the regularity of the evolving networks needed to prove Theorem 5.4.
Lemma 5.2.
Given a sequence of smooth curvature flows of networks
Proof.
See [20, Lemma 8.18]. ∎
Lemma 5.3.
Let
Proof.
One gets the result repeating the arguments presented in [20, Lemma 9.1]. ∎
Theorem 5.4.
Let
for all
Proof.
By hypotheses, we can assume that the sequence of rescaled networks
By means of Lemmas 5.3 and 5.2, we can also assume
that, for
As we the sequence of networks
that is,
for a sequence
Applying now Lemma 4.3 to the flow of networks
We want to see that actually
which is clearly larger than T, as
Choosing
Proof of Theorem 1.1.
Let
a straight line,
a standard triod,
a standard cross.
By White’s local regularity theorem [23],
if the sequence of rescaled curvature flows converges to a straight line, the curvature is
uniformly bounded for
a halfline,
two concurring halflines forming an angle of 120 degrees.
For both these two situation the thesis is obtained as in the case in which
5.3 Limit networks
Now that we know that the curvature is uniformly bounded along the flow, we can show that the
length of the curves of the evolving network converge to a limit as
Lemma 5.5.
Let
Proof.
For every curve
If
Lemma 5.6.
If Conjecture M1 holds, there exist the limits
Proof.
See [20, Lemma 10.7]. ∎
Proposition 5.7.
Let Ω be a bounded, convex and open subset of
either a regular triple junction,
or a 4 - point where the four concurring curves have opposite exterior unit tangent vectors in pairs and form angles of
and the termini on
either external vertices of order one,
or 2 - points where the two concurring curves form an angle of 120 degrees.
Collapse of a curve in the interior and at
an end-point of
Proof.
First of all we note that
by Lemma 5.5
there exist the limits of the lengths of the curves
We reparametrize the curves of
with
Keeping in mind that we are working with the tangential velocity
we can write
This gives
hence, comparing (5.2) and (5.3), we finally deduce that
for every
and by the boundedness of
This implies that
Hence, the family of curves
We deal now with the convergence of the unit tangent vectors. We observe that if we denote with s the arclength parameter, we have
for some constant C, hence, for every sequence of times
Consider
If instead
is zero. Then, for every
Call γ a curve whose end-points are two triple junctions
Now call γ a curve with one internal and one external vertex
To conclude the proof, we must exclude that the length of two adjacent curves
goes to zero as
for every triple junction
for every
which tends to zero, as
Proof of Theorem 1.2.
We only have to show that the curves of
6 An example of a Type-
𝟎
singularity
We have proved that during the flow of regular tree like network the curvature remains bounded, even if the length of a curve goes to zero. We finally show that our main theorem is sharp in the sense that the coalesce of two triple junctions during the flow might actually happen.
Consider an initial regular smooth network
The networks
If the triangle given by the two end-points in the upper half plane and the origin
has the angle at the origin greater than 120 degrees, then there exists only one
minimal network
Funding statement: The authors were partially supported by INDAM–GNAMPA project 2020 Minimal clusters and minimal partitions, the first author by the PRIN Project Geometric flows, optimal transport and metric measure structures and the second and third authors by the PRIN Project Variational methods for stationary and evolution problems with singularities and interfaces.
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Articles in the same Issue
- Frontmatter
- Coherent categorification of quantum loop algebras: The SL(2) case
- Stable 𝔸1-connectivity over a base
- Simple weight modules with finite weight multiplicities over the Lie algebra of polynomial vector fields
- Patchworking the Log-critical locus of planar curves
- Non-pluripolar energy and the complex Monge–Ampère operator
- Type-0 singularities in the network flow – Evolution of trees
- Categorical crystals for quantum affine algebras
- Triviality of the Hecke action on ordinary Drinfeld cuspforms of level Γ1(tn )
- Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence
Articles in the same Issue
- Frontmatter
- Coherent categorification of quantum loop algebras: The SL(2) case
- Stable 𝔸1-connectivity over a base
- Simple weight modules with finite weight multiplicities over the Lie algebra of polynomial vector fields
- Patchworking the Log-critical locus of planar curves
- Non-pluripolar energy and the complex Monge–Ampère operator
- Type-0 singularities in the network flow – Evolution of trees
- Categorical crystals for quantum affine algebras
- Triviality of the Hecke action on ordinary Drinfeld cuspforms of level Γ1(tn )
- Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence
