# A local spectral condition for strong compactness with some applications to
bilateral weighted shifts

Miguel Lacruz and María del Pilar Romero de la Rosa

###### Abstract.

An algebra of bounded linear operators on a Banach space is said to be
strongly compact if its unit ball is precompact in the strong operator
topology, and a bounded linear operator on a Banach space is said to be
strongly compact if the algebra with identity generated by the operator is
strongly compact. Our interest in this notion stems from the work of Lomonosov
on the existence of invariant subspaces. We provide a local spectral condition
that is sufficient for a bounded linear operator on a Banach space to be
strongly compact. This condition is then applied to describe a large class of
strongly compact, injective bilateral weighted shifts on Hilbert spaces,
extending earlier work of Fernández-Valles and the first author. Further
applications are also derived, for instance, a strongly compact, invertible
bilateral weighted shift is constructed in such a way that its inverse fails
to be a strongly compact operator.

††2000 Mathematics Subject Classification. Primary 47B07; Secondary 47B37,
47L10.††Keywords and phrases: Strongly compact operator, local spectral
condition, bilateral weighted shift.††This work was partially supported by
Ministerio de Ciencia e Innovación under project MTM 2009-08934, and by Junta
de Andalucía under grant FQM-3737 .

##  1\. Introduction

Let $\mathcal{B}(X)$ denote the algebra of bounded linear operators on a
complex Banach space $X$. A subalgebra $\mathcal{R}\subseteq\mathcal{B}(X)$ is
said to be strongly compact if its unit ball $\\{R\in\mathcal{R}:\|R\|\leq
1\\}$ is precompact in the strong operator topology, and an operator
$T\in\mathcal{B}(X)$ is said to be strongly compact if the algebra with
identity generated by $T$ is strongly compact.

Our interest in this notion stems from the work of Lomonosov [7] on the
existence of invariant subspaces for essentially normal operators on Hilbert
space. He showed that if $T$ is an essentially normal operator on a Hilbert
space such that both its commutant $\\{T\\}^{\prime}$ and the commutant
$\\{T^{\ast}\\}^{\prime}$ of its adjoint fail to be strongly compact algebras
then $T$ has a nontrivial invariant subspace, and moreover, if both $T$ and
$T^{\ast}$ fail to be strongly compact operators then $T$ has a nontrivial
hyperinvariant subspace.

Lomonosov, Radjavi and Troitsky [8] showed more recently that if $T$ is an
operator such that its commutant $\\{T\\}^{\prime}$ is a localizing algebra
and the commutant $\\{T^{\ast}\\}^{\prime}$ of its adjoint is a strongly
compact algebra, then the adjoint $T^{\ast}$ has an invariant subspace.

A characterization of strongly compact, normal operators was given by
Lomonosov, Rodríguez-Piazza, and the first author [5] by a suitable
application of the spectral theorem. Necessary and sufficient conditions were
also provided for a unilateral weighted shift to be strongly compact in terms
of the sliding products of its weights. Some applications were derived, for
instance, the restriction of a strongly compact operator to an invariant
subspace need not be a strongly compact operator.

Prunaru [11] used a particular case of a more general result obtained by
Lomonosov [7] to show that for a pure hyponormal, essentially normal operator
$T$ on a Hilbert space, the commutant $\\{T^{\ast}\\}^{\prime}$ of its adjoint
is a strongly compact algebra.

Rodríguez-Piazza and the first author [6] showed that the position operator on
the space of square integrable functions with respect to a finite measure of
compact support is strongly compact if and only if the restriction of the
measure to the exterior boundary of its support is purely atomic. (A similar
result was obtained earlier on by Froelich and Marsalli [4] within the
framework of function algebras.) Further applications were derived, for
instance, the weakly closed algebra generated by a strongly compact normal
operator need not be a strongly compact algebra.

A classification of operator algebras was provided by Marsalli [9]. He gave
some sufficient conditions for an algebra of operators to be strongly compact.
He showed that if an operator has a set of eigenvectors that generate a dense
linear manifold, then the operator is strongly compact, and moreover, if all
the corresponding eigenvalues have finite multiplicity, then the commutant of
the operator is strongly compact. Quite recently, Fernández-Valles and the
first author [3] applied this condition to test strong compactness for several
classes of operators, namely, Cesàro operators, bilateral weighted shifts, and
composition operators.

A condition of a different nature seems to be needed to prove strong
compactness in absence of eigenvalues. The purpose of this paper is to provide
a local spectral condition that is sufficient for an operator on a Banach
space to be strongly compact. The condition requires from the operator that
the origin must lie in the interior of its full spectrum and that there must
be a spanning set of vectors where the local spectral radius is smaller than
the distance from the origin to the boundary of the full spectrum. This
condition can be applied to operators with no eigenvalues and it fits like a
glove to bilateral weighted shifts.

As an application, we describe a large class of strongly compact, bilateral
weighted shifts on Hilbert spaces, completing the work of Fernández-Valles and
the first author [3]. Then, we restrict our attention to invertible bilateral
weighted shifts and we derive a condition for such operators to have a
strongly compact inverse. Next, we show that if an invertible bilateral
weighted shift and its inverse are strongly compact operators then the algebra
generated by both of them is strongly compact. Finally, we give an example of
a strongly compact, invertible bilateral weighted shift whose inverse fails to
be strongly compact.

##  2\. A local spectral condition for strong compactness

Let $T$ denote an operator on a complex Banach space $X$ and let $\sigma(T)$
denote its spectrum. The spectral radius of $T$ is defined as
$r(T)=\max\\{|\lambda|:\lambda\in\sigma(T)\\}$. The spectral radius formula
provides the alternative expression

$r(T)=\lim_{n\rightarrow\infty}\|T^{n}\|^{1/n}.$

The local spectral radius of $T$ at a vector $x\in X$ is defined as

$r(x,T)=\limsup_{n\rightarrow\infty}\|T^{n}x\|^{1/n}.$

It is clear that $r(x,T)\leq r(T)$ for every $x\in X$. Daneš [2] showed that
the set of vectors $x\in X$ that satisfy the strict inequality $r(x,T)<r(T)$
is of the first category. See also the paper of Müller [10] for more
information on this remarkable fact and other related results.

Let $G$ denote the unbounded connected component of
$\mathbb{C}\backslash\sigma(T)$. The full spectrum of $T$ is the set
$\eta(\sigma(T))=\mathbb{C}\backslash G$. It is plain that
$\sigma(T)\subseteq\eta(\sigma(T))$ and that
$\mathbb{C}\backslash\eta(\sigma(T))$ is connected. Also, it follows from the
maximum modulus principle that if $p$ is any polynomial then

$\max\\{\,|p(\lambda)|:\lambda\in\sigma(T)\\}=\max\\{\,|p(\lambda)|:\lambda\in\eta(\sigma(T))\\}.$

Recall that $S\subseteq X$ is a spanning set if $S$ generates a dense linear
subspace of $X$. It turns out that a subalgebra
${\mathcal{R}}\subseteq{\mathcal{B}}(X)$ is strongly compact if and only if
there is a spanning set $S\subseteq X$ such that
$\\{Rx:R\in{\mathcal{R}},\,\|R\|\leq 1\\}$ is precompact for every $x\in S$.
The main result on this paper relies on this simple fact. See the paper of
Lomonosov, Rodríguez-Piazza and the first author [5] for a proof of it, and
see also the paper of Marsalli [9] for the same result under a slightly
different formulation.

######  Theorem 1.

Let $T$ be an operator on a complex Banach space $X$ with $0\in{\rm
int\,}\eta(\sigma(T))$ and suppose that there is a spanning set $S\subseteq X$
such that $r(x,T)<d(0,\partial\eta(\sigma(T)))$ for every $x\in S$. Then the
operator $T$ is strongly compact.

###### Proof.

First, consider the positive distance
$d=d(0,\partial\eta(\sigma(T)))=\min\\{|\lambda|:\lambda\in\partial\eta(\sigma(T))\\}$.
Since $0\in{\rm int\,}\eta(\sigma(T))$, we have
$\\{\lambda\in\mathbb{C}:|\lambda|\leq d\\}\subseteq\eta(\sigma(T))$. It
follows from the above remarks and the spectral mapping theorem that for any
polynomial $p$ we have

$\displaystyle\max\\{|p(\lambda)|:|\lambda|=d\\}$ $\displaystyle\leq$
$\displaystyle\max\\{\,|p(\lambda)|:\lambda\in\eta(\sigma(T))\\}$
$\displaystyle=$ $\displaystyle\max\\{\,|p(\lambda)|:\lambda\in\sigma(T)\\}$
$\displaystyle=$ $\displaystyle r(p(T))\leq\|p(T)\|.$

This inequality allows us to control the size of the coefficients for a
complex polynomial $p(\lambda)=a_{0}+a_{1}\lambda+\cdots+a_{m}\lambda^{m}$.
Indeed, using Cauchy’s integral formula for the derivatives of order
$n=0,1,\ldots,m$ gives

$a_{n}=\frac{p^{(n)}(0)}{n!}=\frac{1}{2\pi
i}\int_{|\lambda|=d}\,\frac{p(\lambda)}{\lambda^{n+1}}\,d\lambda,$

and from here we get the estimate

$|a_{n}|\leq\frac{\|p(T)\|}{d^{n}}\,.$

Take a vector $x\in S$. We need to show that the set $\\{p(T)x:\textrm{$p$ is
a polynomial},\,\|p(T)\|\leq 1\\}$ is precompact. Choose some $c>0$ with
$r(x,T)<c<d$. Since $r(x,T)=\limsup\|T^{n}x\|^{1/n}$ as $n\rightarrow\infty$,
there is an $n_{0}$ such that $\|T^{n}x\|\leq c^{n}$ whenever $n\geq n_{0}$.
Let $\varepsilon>0$ and choose an $n_{1}\geq n_{0}$ such that

$\sum_{n=n_{1}+1}^{\infty}\left(\frac{c}{d}\right)^{n}<\varepsilon.$

Finally, consider the finite dimensional subspace $F={\rm
span\,}\\{x,Tx,\ldots,T^{n_{1}}x\\}$. We claim that $p(T)x\in F+\varepsilon
B_{X}$. This is obvious if $m\leq n_{1}$. Otherwise, $p(T)x$ can be expressed
as the sum of two terms,

$p(T)x=\sum_{n=0}^{n_{1}}a_{n}z^{n}+\sum_{n=n_{1}+1}^{m}a_{n}z^{n}.$

The first term belongs to $F$, while the second term satisfies the estimate

$\left\|\sum_{n=n_{1}+1}^{m}a_{n}T^{n}x\right\|\leq\sum_{n=n_{1}+1}^{m}|a_{n}|\cdot\|T^{n}x\|\leq\sum_{n=n_{1}+1}^{m}\left(\frac{c}{d}\right)^{n}<\varepsilon,$

which completes the proof of our claim. ∎

It is easy to see that strong compactness is preserved under similarities,
that is, if $T$ is a strongly compact operator, $C$ is invertible, and
$T=C^{-1}RC$ then $R$ is strongly compact. Notice that the assumption in
Theorem 1 is also preserved under similarities. More precisely, if $T$
satisfies the condition in Theorem 1, $C$ is invertible, and $T=C^{-1}RC$,
then $\sigma(T)=\sigma(R)$, $\\{Cx:x\in S\\}$ is a spanning set, and
$r(Cx,R)\leq r(x,T)<d(0,\partial\eta(\sigma(T)))=d(0,\partial\eta(\sigma(R)))$
for every $x\in S$, so that $R$ also satisfies the condition in Theorem 1.

We now recall a couple of definitions from general operator theory. If $T$ is
any operator on a Banach space $X$ then the lower bound of $T$ is defined as

$m(T)=\inf\\{\|Tx\|:x\in X,\|x\|\leq 1\\}.$

We also consider the quantity

$r_{1}(T)=\sup_{n\geq
1}m(T^{n})^{1/n}=\lim_{n\rightarrow\infty}m(T^{n})^{1/n}.$

It turns out that if $T$ is invertible then $\|T^{-1}\|=m(T)^{-1}$ and
$r(T^{-1})=r_{1}(T)^{-1}$.

##  3\. Applications to bilateral weighted shifts

Let $W$ be an injective bilateral weighted shift on an infinite dimensional,
separable complex Hilbert space $H$, that is,

$We_{n}=w_{n}e_{n+1},$

where $(e_{n})$ is an orthonormal basis of $H$, the weight sequence $(w_{n})$
is bounded, $n$ runs through the integers, and $w_{n}\neq 0$ for every $n$. We
refer to the survey by Allen Shields [12] for information on the spectral
parts of weighted shifts. See also the paper of Bourhim [1] for local spectral
properties of weighted shifts. Now consider the quantities

$r^{-}(W)=\lim_{n\rightarrow\infty}\sup_{k>0}|w_{-n-k}\cdots
w_{-k+1}|^{1/n},\hskip 30.00005ptr^{+}(W)=\lim_{n\rightarrow\infty}\sup_{k\geq
0}|w_{k}\cdots w_{n+k-1}|^{1/n},$
$r_{1}^{-}(W)=\lim_{n\rightarrow\infty}\inf_{k>0}|w_{-n-k}\cdots
w_{-k+1}|^{1/n},\hskip
30.00005ptr_{1}^{+}(W)=\lim_{n\rightarrow\infty}\inf_{k\geq 0}|w_{k}\cdots
w_{n+k-1}|^{1/n},$ $r_{2}^{-}(W)=\liminf_{n\rightarrow\infty}|w_{-1}\cdots
w_{-n}|^{1/n},\hskip
30.00005ptr_{2}^{+}(W)=\liminf_{n\rightarrow\infty}|w_{0}\cdots
w_{n-1}|^{1/n},$ $r_{3}^{-}(W)=\limsup_{n\rightarrow\infty}|w_{-1}\cdots
w_{-n}|^{1/n},\hskip
30.00005ptr_{3}^{+}(W)=\limsup_{n\rightarrow\infty}|w_{0}\cdots
w_{n-1}|^{1/n}.$

Then the following relationships are fulfilled

$r_{1}^{-}(W)\leq r_{2}^{-}(W)\leq r_{3}^{-}(W)\leq r^{-}(W),\hskip
30.00005ptr_{1}^{+}(W)\leq r_{2}^{+}(W)\leq r_{3}^{+}(W)\leq r^{+}(W),$
$r(W)=\max\\{r^{-}(W),r^{+}(W)\\},\hskip
30.00005ptr_{1}(W)=\min\\{r_{1}^{-}(W),r_{1}^{+}(W)\\}.$

It was shown by Fernández-Valles and the first author [3] that if a bilateral
weighted shift $W$ satisfies the inequality $r_{3}^{+}(W)<r_{2}^{-}(W)$ then
its commutant $\\{W\\}^{\prime}$ is strongly compact, and it remained an open
question whether or not $W$ is strongly compact when $r_{2}^{-}(W)\leq
r_{3}^{+}(W)$. The following result provides an affirmative answer to this
question for quite a large class of bilateral weighted shifts.

######  Theorem 2.

If a bilateral weighted shift $W$ on an infinite dimensional, separable
complex Hilbert space satisfies the inequality $r_{3}^{+}(W)<r(W)$ then $W$ is
a strongly compact operator.

###### Proof.

The full spectrum of $W$ is the closed disk of radius $r(W)$ centered at the
origin and the orthornormal basis $(e_{k})$ is a spanning subset of $H.$ Thus,
it suffices to show that $r(e_{k},W)=r_{3}^{+}(W)$ for each $k$. This is
trivial when $k=0$ because $\|W^{n}e_{0}\|=|w_{0}\cdots w_{n-1}|$ for every
$n\geq 1,$ so that

$r(e_{0},W)=\limsup_{n\rightarrow\infty}\|W^{n}e_{0}\|^{1/n}=\limsup_{n\rightarrow\infty}|w_{0}\cdots
w_{n-1}|^{1/n}=r_{3}^{+}(W).$

Then, suppose that $k\neq 0$ and notice that $\|W^{n}e_{k}\|=|w_{k}\cdots
w_{n+k-1}|$ for every $n\geq 1.$ Now, there are two possibilities. On the one
hand, if $k>0$ then

$\|W^{n}e_{k}\|=\frac{|w_{0}\cdots w_{n+k-1}|}{|w_{0}\cdots
w_{k-1}|}=\frac{\|W^{n+k}e_{0}\|}{\|W^{k}e_{0}\|},$

so that

$\displaystyle r(e_{k},W)$ $\displaystyle=$
$\displaystyle\limsup_{n\rightarrow\infty}\|W^{n}e_{k}\|^{1/n}=\limsup_{n\rightarrow\infty}\left(\frac{\|W^{n+k}e_{0}\|}{\|W^{k}e_{0}\|}\right)^{1/n}$
$\displaystyle=$
$\displaystyle\limsup_{n\rightarrow\infty}\|W^{n+k}e_{0}\|^{1/n}=\limsup_{n\rightarrow\infty}\left(\|W^{n+k}e_{0}\|^{1/(n+k)}\right)^{(n+k)/n}=r(e_{0},W).$

On the other hand, if $k<0$ and $n>-k$ then

$\|W^{n}e_{k}\|=|w_{k}\cdots w_{-1}|\cdot|w_{0}\cdots
w_{n+k-1}|=\|W^{-k}e_{k}\|\cdot\|W^{n+k}e_{0}\|,$

so that

$\displaystyle r(e_{k},W)$ $\displaystyle=$
$\displaystyle\limsup_{n\rightarrow\infty}\|W^{n}e_{k}\|^{1/n}=\limsup_{n\rightarrow\infty}\left(\|W^{-k}e_{k}\|\cdot\|W^{n+k}e_{0}\|\right)^{1/n}$
$\displaystyle=$
$\displaystyle\limsup_{n\rightarrow\infty}\|W^{n+k}e_{0}\|^{1/n}=\limsup_{n\rightarrow\infty}\left(\|W^{n+k}e_{0}\|^{1/n+k}\right)^{(n+k)/n}=r(e_{0},W),$

as we wanted. ∎

######  Corollary 3.

Let $W$ be an invertible bilateral weighted shift on an infinite dimensional,
separable complex Hilbert space. If $W$ satisfies the inequality
$r_{1}(W)<r_{2}^{-}(W)$ then its inverse $W^{-1}$ is a strongly compact
operator.

###### Proof.

If $W$ is an invertible bilateral weighted shift with sequence of weights
$(w_{n})$ then

$W^{-1}e_{n}=\frac{1}{w_{n-1}}e_{n-1}.$

Consider the unitary operator $U$ defined by $Ue_{n}=e_{-n}$, so that
$U^{\ast}=U=U^{-1}$. Now define $V=U^{\ast}W^{-1}U$ and notice that $V$ is
another bilateral weighted shift, namely, $Ve_{n}=v_{n}e_{n+1}$, where the
sequence of weights $(v_{n})$ is given by

$v_{n}=\frac{1}{w_{-(n+1)}}.$

According with the remarks at the end of the proof of Theorem 1, strong
compactness is preserved under similarities. Hence, it suffices to show that
$V$ is strongly compact. Then, a quick computation yields
$r_{3}^{+}(V)=r_{2}^{-}(W)^{-1}<r_{1}(W)^{-1}=r(V).$ ∎

######  Theorem 4.

Let $W$ be an invertible bilateral weighted shift on an infinite dimensional,
separable complex Hilbert space. If both $W$ and $W^{-1}$ are strongly compact
then the algebra generated by $W$ and $W^{-1}$ is strongly compact.

###### Proof.

It suffices to show that the set

$C_{k}=\\{p(W)e_{k}+q(W^{-1})e_{k}:p,q\textrm{ are polynomials,
}q(0)=0,\|p(W)+q(W^{-1})\|\leq 1\\}$

is precompact for every integer $k$. Since $W$ and $W^{-1}$ are both strongly
compact, the sets

$\displaystyle C_{k}^{+}$ $\displaystyle=$
$\displaystyle\\{p(W)e_{k}:\textrm{$p$ is a polynomial},\,\|p(W)\|\leq 1\\}$
$\displaystyle C_{k}^{-}$ $\displaystyle=$
$\displaystyle\\{q(W^{-1})e_{k}:\textrm{$q$ is a
polynomial},\,q(0)=0,\,\|q(W^{-1})\|\leq 1\\}$

are both precompact. Notice that for every pair of polynomials $p,q$ with
$q(0)=0$ we have $\|p(W)e_{k}\|\leq\|p(W)e_{k}+q(W^{-1})e_{k}\|$ and
$\|q(W^{-1})e_{k}\|\leq\|p(W)e_{k}+q(W^{-1})e_{k}\|$. Now it follows that
$C_{k}\subseteq C_{k}^{+}+C_{k}^{-}$ so that $C_{k}$ is precompact, as we
wanted. ∎

######  Corollary 5.

Let $W$ be an invertible bilateral weighted shift on an infinite dimensional,
separable complex Hilbert space. If both inequalities $r_{3}^{+}(W)<r(W)$ and
$r_{1}(W)<r_{2}^{-}(W)$ are fulfilled then the algebra generated by $W$ and
$W^{-1}$ is strongly compact.

###### Proof.

Since $r_{3}^{+}(W)<r(W)$, it follows from Theorem 2 that $W$ is strongly
compact, and since $r_{1}(W)<r_{2}^{-}(W)$, it follows from Corollary 3 that
$W^{-1}$ is also strongly compact. Hence, the desired result is a consequence
of Theorem 4. ∎

Example. Consider a bilateral weighted shift $W$ whose sequence of weights
$(w_{n})$ satisfies the conditions $1\leq|w_{n}|\leq 2$ for each $n\geq 0$ and
$w_{n}=1$ for each $n<0$. It is plain that such a weighted shift is bounded
and invertible, and that $\|W^{-1}\|=1$. We claim that $W^{-1}$ fails to be
strongly compact, for otherwise the sequence $e_{-n}=W^{-n}e_{0}$ would have a
norm convergent subsequence as $n\rightarrow\infty$, a contradiction. Then,
the sequence of weights can be chosen in such a way that

$\limsup_{n\rightarrow\infty}|w_{0}\cdots w_{n-1}|^{1/n}=1\hskip
20.00003pt\text{ and }\hskip 20.00003pt\lim_{n\rightarrow\infty}\sup_{k\geq
0}|w_{k}\cdots w_{n+k-1}|^{1/n}=2.$

There are many possible choices for such a sequence of weights, for instance,
if we set $w_{k}=2$ whenever $2^{n}\leq k\leq 2^{n}+n$ and $w_{k}=1$
otherwise. Therefore, $r_{3}^{+}(W)=1$ and $r(W)=r^{+}(W)=2$. It follows from
Theorem 2 that $W$ is a strongly compact operator. Summarizing, we constructed
a strongly compact bilateral weighted shift whose inverse fails to be strongly
compact. Acknowledgement. We are indebted to Fernando León-Saavedra for his
kind invitation to visit Universidad de Cádiz at Jerez de la Frontera during
the preparation of this manuscript. We had many conversations that made
possible the present work. We would like to express our sincere gratitude for
his generosity and his hospitality.

## References

  * [1] A. Bourhim, ‘On the local spectral properties of weighted shift operators’, Studia Math. 163 (2004), no. 1, 41–69.
  * [2] J. Daneš, ‘On local spectral radius’, Časopis Pěst. Mat. 112 (1987) 177 –187.
  * [3] A. Fernández-Valles M. Lacruz, ‘Spectral conditions for strong compactness’, preprint.
  * [4] J. Froelich and M. Marsalli, ‘Operator semigroups, flows, and their invariant sets’, J. Funct. Anal. 115 (1993), no. 2, 454–479.
  * [5] M. Lacruz, V.I. Lomonosov L. Rodríguez-Piazza, ‘Strongly compact algebras’, Rev. R. Acad. Cien. Serie A. Mat. 100 (2006) 191–207.
  * [6] M. Lacruz, L. Rodríguez-Piazza, ‘Strongly compact normal operators’, Proc. Amer. Math. Soc. 137 (2009), no. 8, 2623–2630.
  * [7] V.I. Lomonosov, ‘A construction of an intertwining operator’, Funktsional. Anal. i Prilozhen. 14 (1980) 67–68.
  * [8] V.I. Lomonosov, H. Radjavi V.G. Troitsky, ‘Sesquitransitive and localizing operator algebras’, Integr. equ. oper. theory 60 (2008) 405 –418.
  * [9] M. Marsalli, ‘A classification of operator algebras’, J. Operator Theory 24 (1990) 155–163.
  * [10] V. Müller, ‘Orbits, weak orbits and local capacity of operators’, Integr. equ. oper. theory 41 (2001) 230 –253.
  * [11] B. Prunaru, ‘On the commutant of hyponormal operators’, Proc. Amer. Math. Soc. 124 (1996) 3411–3412.
  * [12] A.L. Shields,‘Weighted shift operators and analytic function theory’, Topics in operator theory, Math. Surveys Amer. Math. Soc. 13 (1974) 49–128.

Miguel Lacruz |  | María del Pilar Romero de la Rosa
---|---|---
Departamento de Análisis |  | Departamento de Matemáticas
Universidad de Sevilla |  | Universidad de Cádiz
Avenida Reina Mercedes |  | Avenida de la Universidad
41012 Seville (SPAIN) |  | 11402 Jerez de la Frontera (SPAIN)
<EMAIL_ADDRESS>|  |<EMAIL_ADDRESS>


# On the uniqueness of nondegenerate blowups for the motion by curvature of
networks

Alessandra Pluda<EMAIL_ADDRESS>Dipartimento di Matematica,
Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy. Marco
Pozzetta<EMAIL_ADDRESS>Dipartimento di Matematica e
Applicazioni, Università di Napoli Federico II, Via Cintia, Monte S. Angelo
80126 Napoli, Italy

###### Abstract

In this note we prove uniqueness of nondegenerate compact blowups for the
motion by curvature of planar networks. The proof follows ideas introduced in
[18] for the study of stability properties of critical points of the length
functional and it is based on the application of a Łojasiewicz–Simon gradient
inequality.

Mathematics Subject Classification (2020): 53E10, 53A04, 46N20, 35B44.

##  1 Introduction

If $G$ is an abstract one-dimensional graph, see 2.1, a network
$\Gamma:G\to\mathbb{R}^{2}$ is, roughly speaking, an embedding of $G$ in
$\mathbb{R}^{2}$ of class $H^{2}$. Topologically, we only consider graphs
whose junctions have order three, and we will be mainly interested in regular
networks, i.e., networks such that the tangent vectors of the immersions given
by $\Gamma$ forms equal angles, equal to $\tfrac{2}{3}\pi$, at any junction. A
motion by curvature $\Gamma_{t}:G\to\mathbb{R}^{2}$ is a smooth one-parameter
family of networks such that the normal velocity of the evolution is equal to
the curvature of the immersions along each edge, see 2.3. The motion by
curvature of networks generalizes the one-dimensional mean curvature flow,
called curve shortening flow, to singular one-dimensional objects given by
planar networks. The basic theory concerning short and long time behavior of
this geometric flow can be found in [2, 16, 15, 9]. Just like classical mean
curvature flow, the motion by curvature may develop singularities determined
by the fact the either the length of some edge goes to zero or the
$L^{2}$-norm of the curvature diverges in finite time. A great deal of work
has been done to understand the nature of these singularities and to define
the flow past singularities [16, 13, 15, 1, 11, 12, 14, 4, 3], combining a
bunch of different techniques from analysis and geometry.

A nowadays standard procedure employed to analyze a singularity for a flow
$\Gamma_{t}:G\to\mathbb{R}^{2}$ is to perform a _blowup_ of the motion.
Classically, if $\Gamma_{t}$ is defined on times $[0,T)$ and
$x_{0}\in\mathbb{R}^{2}$ is fixed, this means to consider sequences of
parabolically rescaled motions by curvature
$\Gamma^{\mu_{j}}_{t}\vcentcolon=\mu_{j}\big{(}\Gamma_{-\mu_{j}^{-2}t+T}-x_{0}\big{)}:G\to\mathbb{R}^{2}$
and to pass to the limit in space-time as $\mu_{j}\to+\infty$. Up to
subsequence, one shows that there exists such a limit $\Gamma^{\infty}_{t}$,
which is called tangent flow, and it is a homothetically shrinking motion by
curvature, see [15, Section 8]. The shape of $\Gamma^{\infty}_{t}$ at any time
identifies the motion by curvature $\Gamma_{t}$ at infinitesimal scales around
$x_{0}$ as $t$ approaches $T$. However, the construction may clearly depend on
the choice of the sequence $\mu_{j}$, and it is today an open problem to prove
uniqueness of blowups, i.e., independence of the sequence $\mu_{j}$, for a
motion by curvature, or for the mean curvature flow, in full generality. Some
of the most influential contributions to the uniqueness of blowups for mean
curvature flow are contained in [20], for the case of compact blowups, [8],
for the case of cylindrical blowups, [7], for the case of conical blowups.
An equivalent way for proving uniqueness of blowups, employed also in the
above mentioned works, is to prove that the so-called rescaled motion by
curvature
$\widetilde{\Gamma}_{\mathfrak{t}}\vcentcolon=\big{(}2e^{-\mathfrak{t}}\big{)}^{-\frac{1}{2}}\big{(}\Gamma_{T-e^{-\mathfrak{t}}}-x_{0}\big{)}:G\to\mathbb{R}^{2}$
has a full limit as $\mathfrak{t}\to+\infty$, see Section 2.2. As in the case
of tangent flows, along sequences $\mathfrak{t}_{n}\to+\infty$, employing the
celebrated Huisken’s monotonicity formula one always finds limits of
$\widetilde{\Gamma}_{\mathfrak{t}_{n}}$, but the limit may depend on the
diverging sequence of times.

In this note, we prove that if one of such sequential limits $\Gamma_{*}$ of
$\widetilde{\Gamma}_{\mathfrak{t}_{n}}$ is nondegenerate, of multiplicity one,
and compact, encoded in the fact that $\Gamma_{*}:G\to\mathbb{R}^{2}$ is a
regular network for the same graph $G$ which is the domain of the original
flow $\Gamma_{t}:G\to\mathbb{R}^{2}$, then it is unique. For the precise
definitions of the terminology used in the next statement, we refer to 2.1,
2.3, 2.4, and 2.6 below.

######  Theorem 1.1.

Let $\Gamma_{t}:G\to\mathbb{R}^{2}$ be a motion by curvature defined on times
$t\in[0,T)$ with $T<+\infty$, where $G$ is a graph without endpoints.

If $\Gamma_{*}:G\to\mathbb{R}^{2}$ is a nonempty nondegenerate blowup of
$\Gamma_{t}$ at some $x_{0}\in\mathbb{R}^{2}$ and time $T$, then $\Gamma_{*}$
is the unique blowup of $\Gamma_{t}$ at point $x_{0}$ and time $T$.

More precisely, the following two (equivalent) statements hold true.

  1. 1.

The rescaled motion by curvature $\widetilde{\Gamma}_{t}$ at point $x_{0}$ and
time $T$ smoothly converges to $\Gamma_{*}$ as $t\to+\infty$, up to
reparametrizations.

  2. 2.

For any diverging sequence $\mu_{j}\nearrow+\infty$, the sequence of
parabolically rescaled motions by curvature $\Gamma^{\mu_{j}}_{t}$ converges
locally smoothly in space-time to the homothetically shrinking motion by
curvature $\Gamma^{\infty}_{t}$ such that
$\Gamma^{\infty}_{-\frac{1}{2}}=\Gamma_{*}$.

We remark that the above uniqueness theorem holds true regardless of
assumptions on whether $T$ is a singularity time for $\Gamma_{t}$ of Type I or
Type II, see [15, remark 8.21]. On the other hand, a necessary consequence of
the uniqueness result in 1.1 is the fact that if $\Gamma_{t}$ develops a
singularity at time $T$ and a blowup at $T$ is a nondegenerate network
$\Gamma_{*}:G\to\mathbb{R}^{2}$, then such singularity is, in fact, a Type I
singularity (see [15, Remark 8.24]).

1.1 is a consequence of 3.9 below, which proves smooth convergence of a
rescaled motion $\widetilde{\Gamma}_{\mathfrak{t}}$. The great advantage of
working with the flow $\widetilde{\Gamma}_{\mathfrak{t}}$ is that such
evolution turns out to be the $L^{2}$-gradient flow of the _Gaussian energy_
defined by

$\mathcal{G}(\Gamma)\vcentcolon=\sum_{i}\int_{0}^{1}e^{-|\gamma^{i}(x)|^{2}/2}\,\mathrm{d}s,$

for any network $\Gamma:G\to\mathbb{R}^{2}$. With the obvious meaning of
symbols $\sum_{i}\int_{\gamma^{i}}$ and $\int_{\Gamma}$, we shall equivalently
write
$\mathcal{G}(\Gamma)=\sum_{i}\int_{\gamma^{i}}e^{-|p|^{2}/2}=\int_{\Gamma}e^{-|p|^{2}/2}$.
We recall that networks that are critical points of $\mathcal{G}$ are
shrinkers, i.e., the motion by curvature starting from such networks evolves
homothetically contracting, see Section 2.
The gradient flow structure permits to apply a general strategy for proving
full convergence of a flow out of its sequential convergence along sequences
of times, see [19]. Such strategy is based on the application of a
Łojasiewicz–Simon gradient inequality [21, 5, 6] derived on the energy
functional $\mathcal{G}$. We derive such inequality in 3.8 below.

The main difficulty here is that such an inequality would readily follow by
functional analytic results [5] relying on the possibility of parametrizing
networks as normal graphs over fixed critical points of $\mathcal{G}$.
However, the nonsmooth structure of networks necessarily implies technical
complications, since networks close to a fixed one $\Gamma_{*}$ cannot be
written as normal graphs over $\Gamma_{*}$. Hence, in order to perform a graph
parametrization of networks close to $\Gamma_{*}$ we will need to allow for
graphs having both a normal and a tangential component with respect to
$\Gamma_{*}$. The issue of performing such a graph parametrization and at the
same time of achieving the linear functional analytic setup needed for
deriving the Łojasiewicz–Simon inequality was investigated in [18] for the
study of stability properties networks that are critical points of the length
functional. We shall recall from [18] the tools needed here at the beginning
of Section 3, and we refer to [18] for a more detailed account on the method.

Organization. In Section 2 we recall basic definitions on networks, motion by
curvature, and blowups. In Section 3 we first derive the Łojasiewicz–Simon
inequality at critical points of $\mathcal{G}$, 3.8, and then we prove
convergence of the rescaled motion by curvature, 3.9, implying the main result
1.1.

##  2 Preliminaries

###  2.1 Networks

For a regular curve $\gamma:[0,1]\to\mathbb{R}^{2}$ of class $H^{2}$, we
define the tangent vector
$\tau_{\gamma}\vcentcolon=\frac{\gamma^{\prime}}{|\gamma^{\prime}|}$, and the
normal vector $\nu_{\gamma}\vcentcolon={\rm R}(\tau_{\gamma})$, where ${\rm
R}$ denotes counterclockwise rotation of $\tfrac{\pi}{2}$. We define
$\,\mathrm{d}s_{\gamma}\vcentcolon=|\gamma^{\prime}|\,\mathrm{d}x$ the
arclength element and
$\partial_{s}\vcentcolon=|\gamma^{\prime}|^{-1}\partial_{x}$ the arclength
derivative. The curvature of $\gamma$ is the vector

$\boldsymbol{k}_{\gamma}\vcentcolon=\partial_{s}^{2}\gamma.$

We shall usually drop the subscript $\gamma$ when there is no risk of
confusion.

######  Definition 2.1 (Networks).

Fix $N\in\mathbb{N}$ and let $i\in\\{1,\ldots,N\\}$,
$E^{i}:=[0,1]\times\\{i\\}$, $E:=\bigcup_{i=1}^{N}E^{i}$ and
$V:=\bigcup_{i=1}^{N}\\{0,1\\}\times\\{i\\}$.

Let $\sim$ be an equivalence relation that identifies points of $V$. Let
$G\vcentcolon=E\,/\,\sim$ be the graph given by the topological quotient space
of $E$ induced by $\sim$. Assume that $G$ is connected.

Denoting by $\pi:E\to G$ the projection onto the quotient, an _endpoint_ is a
point $p\in G$ such that $\pi^{-1}(p)\subset V$ and it is a singleton, a
_junction_ is a point $m\in G$ such that $\pi^{-1}(m)\subset V$ and it is not
a singleton.

For any $(e^{i},i)\in E^{i}$ such that $\pi(e^{i},i)$ is a junction, we assume
that there exist exactly two boundary points $(e^{j},j)\in E^{j},(e^{k},k)\in
E^{k}$, for $j\neq k$ distinct from $i$, such that
$\pi(e^{i},i)=\pi(e^{j},j)=\pi(e^{k},k)$. A graph with such property is said
to be _regular_.

We denote by $J_{G}$ and $P_{G}$ the set of junctions and endpoints,
respectively.

A _network_ is a continuous map $\Gamma:G\to\mathbb{R}^{2}$ such that
$\Gamma|_{E^{i}}$ is either a $C^{1}$-embedding or a constant map. We say that
$\Gamma$ is of class $C^{k}$ or smooth, if any embedding $\Gamma|_{E^{i}}$ is
of class $C^{k}$ or $C^{\infty}$, respectively.

The _inner tangent vector_ of a regular curve
$\gamma^{i}\vcentcolon=\Gamma|_{E^{i}}$ at $e\in\\{0,1\\}$ is the vector

$(-1)^{e}\frac{(\gamma^{i})^{\prime}(e)}{|(\gamma^{i})^{\prime}(e)|}.$

A _triple junctions_ (or _nondegenerate_) _network_ is an embedding
$\Gamma:G\to\mathbb{R}^{2}$ such that any restriction $\Gamma|_{E^{i}}$ is
nonconstant.

A network is said to be _regular_ if it is triple junctions and whenever
$\pi(e^{i},i)=\pi(e^{j},j)=\pi(e^{k},k)$ is a junction then any two inner
tangent vectors of $\gamma^{i},\gamma^{j},\gamma^{k}$ at $e^{i},e^{j},e^{k}$,
respectively, form an angle equal to $\tfrac{2}{3}\pi$.

Without loss of generality, if $\Gamma:G\to\mathbb{R}^{2}$ is a network and
$p\in G$ is an endpoint with $\pi^{-1}(p)=(e,i)$, we will assume that $e=1$.

In order to treat blowups of the motion by curvature, we introduce the
following definition of complete network.

######  Definition 2.2 (Complete regular network).

Fix $N\in\mathbb{N}$ and let $i\in\\{1,\ldots,N\\}$. Let
$E^{i}\vcentcolon=I^{i}\times\\{i\\}$, where $I^{i}$ is any of the intervals
$[0,1]$, $(0,1]$, $[1,0)$, or $(0,1)$. Let $E\vcentcolon=\cup_{i}E^{i}$.

Let $\sim$ be an equivalence relation that identifies points on the boundaries
of the $E^{i}$’s. Consider a graph $H\vcentcolon=\cup_{i}E^{i}\,/\,\sim$ and
let $\pi$ be the projection onto the quotient. Assume that $H$ is connected
and that for any $(e^{i},i)\in E^{i}$ there exist exactly two boundary points
$(e^{j},j)\in E^{j},(e^{k},k)\in E^{k}$, for $j\neq k$ distinct from $i$, such
that $\pi(e^{i},i)=\pi(e^{j},j)=\pi(e^{k},k)$. Such a graph is said to be
_open_.

A continuous map $\Gamma:H\to\mathbb{R}^{2}$ such that $\Gamma|_{E^{i}}$ is
either a $C^{1}$-embedding or a constant map is still said to be a _network_.

A _complete regular network_ is an embedding $\Gamma:H\to\mathbb{R}^{2}$ such
that $\Gamma|_{E^{i}}$ is a complete embedding of class $C^{1}$ for any $i$
and whenever $\pi(e^{i},i)=\pi(e^{j},j)=\pi(e^{k},k)$ is a junction then any
two inner tangent vectors of $\gamma^{i},\gamma^{j},\gamma^{k}$ at
$e^{i},e^{j},e^{k}$, respectively, form an angle equal to $\tfrac{2}{3}\pi$.

Whenever a graph $G$ is not explicitly said to be open, then it is assumed to
be as in 2.1; in particular, such a $G$ is compact.

Observe that, by definition, a complete regular network has no endpoints.
Also, a regular network $\Gamma:G\to\mathbb{R}^{2}$ such that $G$ has no
endpoints is automatically a complete regular network. This choice is coherent
with the fact that we shall restrict ourselves to the study of compact
blowups.

###  2.2 Motion by curvature and blowups

######  Definition 2.3 (Motion by curvature).

A smooth one-parameter family of regular networks
$\Gamma_{t}:G\to\mathbb{R}^{2}$, for $t\in[0,T)$, is a motion by curvature if
the parametrizations $\gamma^{i}_{t}$ of $\Gamma_{t}$ satisfy

$\begin{split}\gamma^{i}_{t}(p)=P^{i}&\qquad\forall\,t\in[0,T),\,\,\forall\,p\text{
endpoint of $G$},\\\
\left\langle\partial_{t}\gamma^{i}(t,x),\nu^{i}(t,x)\right\rangle\nu^{i}(t,x)=\boldsymbol{k}^{i}(t,x)&\qquad\forall\,t\in[0,T),\,\,\forall\,x\in(0,1),\end{split}$
(2.1)

for some points $P^{i}\in\mathbb{R}^{2}$. The solution is assumed to be
maximal, i.e., we ask that it does not exists another solution defined on
$[0,\widetilde{T})$ with $\widetilde{T}>T$.

For ease of notation, we shall often write that
$\langle\partial_{t}\Gamma_{t},\nu_{t}\rangle\nu_{t}=\boldsymbol{k}_{t}$,
understanding that the relation holds along any edge $E^{i}$.

Exploiting the analysis carried out in [9], we can assume that the
parametrizations of a motion by curvature evolve according to the _special
flow_ , i.e., the evolution is determined by the non-degenerate parabolic
second order equation

$\partial_{t}\gamma^{i}_{t}=\frac{\partial^{2}_{x}\gamma^{i}_{t}}{|\partial_{x}\gamma^{i}_{t}|^{2}}\,.$
(2.2)

We stress that, for our purposes, it is not restrictive to assume that a
motion by curvature is a smooth evolution of networks, thanks to the result
concerning parabolic regularization of the evolution in [9].

In this paper we are interested in a careful description of the singularities
of the flow, hence we will assume that the flows develops a singulatiry at a
finite time $T<\infty$. In view of [15, Theorem 5.14], this means that either
the inferior limit of the length of at least one curve of the network is zero
as $t\to T^{-}$, or the superior limit of the $L^{2}$-norm of the curvature is
$+\infty$ as $t\to T^{-}$. We remind the reader that there are explicit
example of singularity formations in the network flow, where one curves
vanishes and two triple junctions coalesce, or entire regions disappears (see
for instance [17, 15, 18]).

We introduce now a rescaling procedure first used by Huisken in [10].

######  Definition 2.4 (Rescaled motion by curvature).

If $\Gamma_{t}:G\to\mathbb{R}^{2}$ is a motion by curvature, the _rescaled
motion by curvature at $x_{0}\in\mathbb{R}^{2}$ and time $T$_ is the flow

$\widetilde{\Gamma}_{x_{0}}:[-\tfrac{1}{2}\log T,+\infty)\times
G\to\mathbb{R}^{2}\qquad\qquad\widetilde{\Gamma}_{x_{0}}(\mathfrak{t},x)\vcentcolon=\frac{\Gamma_{t}(x)-x_{0}}{\sqrt{2(T-t)}},$

where $\mathfrak{t}(t)\vcentcolon=-\tfrac{1}{2}\log(T-t)$. We shall denote
$\widetilde{\Gamma}_{\mathfrak{t}}(\cdot)\vcentcolon=\widetilde{\Gamma}_{x_{0}}(\mathfrak{t},\cdot)$,
omitting the dependence on $x_{0}$.

Without loss of generality, in the definition of rescaled motion we shall
always assume that $x_{0}=0$ is the origin of $\mathbb{R}^{2}$.

A direct computation shows that a rescaled motion by curvature evolves
according to

$\partial_{\mathfrak{t}}\widetilde{\Gamma}_{\mathfrak{t}}=\sqrt{2(T-t)}\partial_{t}\Gamma_{t}+\widetilde{\Gamma}_{\mathfrak{t}},$

for $t(\mathfrak{t})=T-e^{-2\mathfrak{t}}$, understanding that the relation
holds along any edge $E^{i}$. In particular normal and tangent vectors of
$\widetilde{\Gamma}_{\mathfrak{t}}$ coincide with the ones of $\Gamma_{t}$,
that is $\widetilde{\nu}_{\mathfrak{t}}(x)=\nu_{t}(x)$ and
$\widetilde{\tau}_{\mathfrak{t}}(x)=\tau_{t}(x)$, and the normal velocity is

$\langle\partial_{\mathfrak{t}}\widetilde{\Gamma}_{\mathfrak{t}},\widetilde{\nu}_{\mathfrak{t}}\rangle\widetilde{\nu}_{\mathfrak{t}}=\widetilde{\boldsymbol{k}}_{\mathfrak{t}}+\langle\widetilde{\Gamma}_{\mathfrak{t}},\widetilde{\nu}_{\mathfrak{t}}\rangle\widetilde{\nu}_{\mathfrak{t}}.$

In view of [15, Section 7.2] and 3.5 below, it turns out that a rescaled
motion by curvature $\widetilde{\Gamma}_{\mathfrak{t}}$ is an $L^{2}$-gradient
flow of the Gaussian energy $\mathcal{G}$. Moreover, a complete regular
network $\Gamma_{*}:H\to\mathbb{R}^{2}$ is critical point for $\mathcal{G}$,
i.e., $\boldsymbol{k}_{*}+\langle\Gamma_{*},\nu_{*}\rangle\nu_{*}=0$, if and
only if $\Gamma_{*}$ is a _shrinker_ , that is, $\Gamma_{*}$ is complete
regular and the motion by curvature starting from $\Gamma_{*}$ evolves
homothetically contracting.

######  Theorem 2.5 ([15, Proposition 8.20, Remark 8.21]).

Let $\widetilde{\Gamma}_{\mathfrak{t}}$ be a rescaled motion by curvature at
the origin $0$ and at maximal time $T$ of existence of a motion by curvature
$\Gamma_{t}$. Assume that $0$ is not an endpoint of the motion $\Gamma_{t}$.
Then the following hold.

  1. 1.

There exists a diverging sequence of times $\mathfrak{t}_{n}\to+\infty$ and a
(possibly empty) open subgraph $H\subset G$ such that: for any $E^{i}\subset
G\setminus H$ and any $R>0$ it holds
$\widetilde{\gamma}^{i}_{\mathfrak{t}_{n}}(E^{i})\cap B_{R}(0)=\emptyset$ for
any $n$ large enough, for any $E^{i}\cap H\subset H$ the curves
$\widetilde{\gamma}^{i}_{\mathfrak{t}_{n}}$ converge in $C^{1}\cap H^{2}$
locally on $E^{i}\cap H$, up to reparametrization, to either a constant curve
or to an embedding of class $H^{2}$.

  2. 2.

If also the sequence $\widetilde{\Gamma}_{\mathfrak{t}_{n}}|_{H}$ converges to
a complete regular network $\Gamma_{*}:H\to\mathbb{R}^{2}$ in $C^{1}\cap
H^{2}$ locally on $H$, up to reparametrizations, then convergence holds in
$C^{k}$ locally on $H$ for any $k\in\mathbb{N}$.

In the notation of the first item in 2.5, the network
$\Gamma_{*}:H\to\mathbb{R}^{2}$ given by the limit immersions
$\Gamma_{*}|_{E^{i}}\vcentcolon=\lim_{n}\widetilde{\gamma}^{i}_{\mathfrak{t}_{n}}$
is said to be a _blowup limit at $0$ and time $T$_ or, in short, _blowup_ of
$\Gamma_{t}$.

If the second item in 2.5 occurs, we say that $\Gamma_{*}$ is a _nondegenerate
blowup of $\Gamma_{t}$ (at $0$ and time $T$)_.

In the second item of 2.5, the requirement that
$\widetilde{\Gamma}_{\mathfrak{t}_{n}}|_{H}$ converges to a complete regular
network $\Gamma_{*}:H\to\mathbb{R}^{2}$ is equivalent to the fact that
$\widetilde{\gamma}^{i}_{\mathfrak{t}_{n}}$ converges to an embedding
$\gamma^{i}_{*}$ for any $E^{i}\cap H\subset H$ up to reparametrizations, and
$\gamma^{i}_{*}(E^{i}\cap H)$ intersects $\gamma^{j}_{*}(E^{j}\cap H)$ for
$i\neq j$ only at junctions, i.e., the limit network has multiplicity one.
We remark that using a modified version of Huisken’s monotonicity formula [10]
one proves that nondegenerate blowups of a motion by curvature are
automatically shrinkers.

An alternative and equivalent way for defining blowups is given by the already
mentioned concept of tangent flow, recalled in the next remark.

######  Remark 2.6 (Parabolic rescaling and tangent flows).

Let $\Gamma_{t}:G\to\mathbb{R}^{2}$ be a motion by curvature defined on the
time interval $[0,T)$. For any $\mu>0$, the _parabolically rescaled motion by
curvature $\Gamma^{\mu}_{t}$ (at point $x_{0}$ and time $T$)_ is the evolution
$\Gamma^{\mu}_{t}:G\to\mathbb{R}^{2}$ given by

$\Gamma^{\mu}_{t}|_{E^{i}}(x)\vcentcolon=\mu\big{(}\Gamma_{\mu^{-2}\,t+T}|_{E^{i}}(x)-x_{0}\big{)},$

for $t\in[-\mu^{2}T,0)$ and any $i=1,\ldots,N$ and $x\in E^{i}$. It is readily
checked that $\Gamma^{\mu}_{t}$ is still a motion by curvature.
Suppose without loss of generality that $x_{0}=0$, then $\Gamma^{\mu}_{t}$ is
related to the rescaled motion by curvature
$\widetilde{\Gamma}_{\mathfrak{t}}$ at time $T$ via the identities

$\Gamma^{\mu}_{t}=\sqrt{-2t}\,\widetilde{\Gamma}_{\log(\mu/\sqrt{-t})},\qquad\widetilde{\Gamma}_{\mathfrak{t}}=\frac{e^{\mathfrak{t}}}{\mu\sqrt{2}}\Gamma^{\mu}_{-\mu^{2}e^{-2\mathfrak{t}}}.$

In particular
$\Gamma^{\mu}_{-\frac{1}{2}}=\widetilde{\Gamma}_{\log(\mu\sqrt{2})}$, whenever
$-\mu^{2}T\leq-\tfrac{1}{2}$.
Consider now a sequence $\mu_{j}\nearrow+\infty$, together with the sequence
of flows $\Gamma^{\mu_{j}}_{t}$. Exploiting the above identities and 2.5, or
(equivalently) [15, Proposition 8.16, Lemma 8.18], one can prove that, up to
extracting a subsequence, there is an open subgraph $H\subset G$ such that:
for any $E^{i}\subset G\setminus H$ and any $R>0$ it holds
$\Gamma^{\mu_{j}}_{t}(E^{i})\cap B_{R}(0)=\emptyset$ for any $t<0$ and $j$
large enough, for any $E^{i}\cap H\subset H$ the curves
$\Gamma^{\mu_{j}}_{t}|_{E^{i}}$ converge in $C^{1}\cap H^{2}$ locally on
$E^{i}\cap H$ to curves $\Gamma^{\infty}_{t}|_{E^{i}}$ for any $t<0$ as
$j\to+\infty$, up to reparametrization, which are either constant or
embeddings of class $H^{2}$. The flow $\Gamma^{\infty}_{t}$ is said to be a
_tangent flow_ to $\Gamma_{t}$.
In case $\Gamma^{\infty}_{t}:H\to\mathbb{R}^{2}$ is regular for some (hence,
any) $t$, then $\Gamma^{\infty}_{t}$ is a homothetically shrinking motion by
curvature and $\Gamma^{\infty}_{-\frac{1}{2}}$ is the limit of the rescaled
motion $\widetilde{\Gamma}_{\mathfrak{t}}$ along the sequence of times
$\mathfrak{t}_{j}=\log(\mu_{j}\sqrt{2})$.
Conversely, if $\Gamma_{*}$ is a sequential limit of
$\widetilde{\Gamma}_{\mathfrak{t}}$ along some sequence of times
$\mathfrak{t}_{n}$ as in 2.5, then there exists the tangent flow
$\Gamma^{\infty}_{t}$ obtained as the limit of the parabolic rescalings
$\Gamma^{\mu_{n}}_{t}$ taking $\mu_{n}=e^{\mathfrak{t}_{n}}/\sqrt{2}$.
Hence blowups of a motion by curvature are equivalently identified by tangent
flows or by sequential limits of the rescaled motion by curvature, see also
[15, Remark 8.27, Remark 7.4].

######  Remark 2.7.

In the notation and hypotheses of 2.5, if item 2 occurs, if also
$\Gamma_{*}(H)$ is compact and nonempty, then $H=G$ is a regular graph without
endpoints and $\Gamma_{*}:G\to\mathbb{R}^{2}$ is a regular network.

##  3 Uniqueness of blowups

We need to recall the setup introduced in [18, Section 3.1] in order to
canonically parametrize networks close in $H^{2}$ to a regular one
$\Gamma_{*}$ suitably as “graphs” over $\Gamma_{*}$.

For a given regular network $\Gamma_{*}$, for parametrizations of the form
$\gamma^{\ell}(x)\vcentcolon=\gamma^{\ell}_{*}(x)+\mathsf{N}^{\ell}(x)\nu^{\ell}_{*}(x)+\mathsf{T}^{\ell}(x)\tau^{\ell}_{*}(x)$,
the next lemma contains sufficient conditions to be imposed on the functions
$\mathsf{N}^{\ell},\mathsf{T}^{\ell}$ so that the resulting parametrizations
$\gamma^{\ell}$ define a triple junctions network. It readily seen that such
conditions are also necessary [18, Lemma 3.1].

######  Lemma 3.1 ([18, Lemma 3.2]).

Let $\Gamma_{*}:G\to\mathbb{R}^{2}$ be a regular network. Then there exists
$\varepsilon_{\Gamma_{*}}>0$ such that for every
$\mathsf{N}^{\ell},\mathsf{T}^{\ell}\in C^{1}([0,1])$, with
$\|\mathsf{N}^{\ell}\|_{C^{1}},\|\mathsf{T}^{\ell}\|_{C^{1}}\leq\varepsilon_{\Gamma_{*}}$
fulfilling at any junction $m=\pi(e^{i},i)=\pi(e^{j},j)=\pi(e^{k},k)$ the
following identities

$\begin{split}&\mathsf{T}^{i}(e^{i})=-\frac{1}{\sqrt{3}}\mathsf{N}^{i}(e^{i})-\frac{2}{\sqrt{3}}(-1)^{e_{i}+e_{j}}\mathsf{N}^{j}(e^{j})\,,\\\
&\mathsf{T}^{j}(e^{j})=\frac{2}{\sqrt{3}}(-1)^{e_{i}+e_{j}}\mathsf{N}^{i}(e^{i})+\frac{1}{\sqrt{3}}\mathsf{N}^{j}(e^{j})\,,\\\
&\mathsf{T}^{k}(e^{k})=-\frac{1}{\sqrt{3}}(-1)^{e_{i}+e_{k}}\mathsf{N}^{i}(e^{i})+\frac{1}{\sqrt{3}}(-1)^{e_{j}+e_{k}}\mathsf{N}^{j}(e^{j})\,,\\\
&(-1)^{e^{i}}\mathsf{N}^{i}(e^{i})+(-1)^{e^{j}}\mathsf{N}^{j}(e^{j})+(-1)^{e^{k}}\mathsf{N}^{k}(e^{k})=0\,,\end{split}$
(3.1)

the maps

$\gamma^{\ell}(x)\vcentcolon=\gamma^{\ell}_{*}(x)+\mathsf{N}^{\ell}(x)\nu^{\ell}_{*}(x)+\mathsf{T}^{\ell}(x)\tau^{\ell}_{*}(x),$

define a triple junctions network.

The above result motivates the following definitions.

######  Definition 3.2.

Let $G$ be a regular graph. At any junction
$m=\pi(e^{i},i)=\pi(e^{j},j)=\pi(e^{k},k)$, if $i<j<k$, we denote by
$L^{i},L^{j},L^{k}$ the linear maps

$\begin{split}&L^{i}(a,b)=-\frac{1}{\sqrt{3}}a-\frac{2}{\sqrt{3}}(-1)^{e_{i}+e_{j}}b\,,\\\
&L^{j}(a,b)=\frac{2}{\sqrt{3}}(-1)^{e_{i}+e_{j}}a+\frac{1}{\sqrt{3}}b\,,\\\
&L^{k}(a,b)=-\frac{1}{\sqrt{3}}(-1)^{e_{i}+e_{k}}a+\frac{1}{\sqrt{3}}(-1)^{e_{j}+e_{k}}b\,.\end{split}$

Moreover, we denote by $I_{m}$ the set of indices $\ell$ such that $E^{\ell}$
has an endpoint at $m$, and we denote by $e^{\ell}_{m}\in\\{0,1\\}$ the
endpoint of $E^{\ell}$ at $m$, for $\ell\in I_{m}$.
Furthermore, for $\ell\in I_{m}$, we denote by $\mathscr{L}^{\ell}_{m}$ the
linear operator $L^{i},L^{j},L^{k}$, depending on whether $\ell$ is the
minimal, intermediate, or maximal index $\ell\in I_{m}$.

We fix a nonincreasing smooth cut-off function

$\chi:[0,1/2]\to[0,1],\quad\chi|_{[0,\frac{1}{8}]}=1,\quad\chi|_{[\frac{3}{8},\frac{1}{2}]}\equiv
0.$

######  Definition 3.3 (Adapted tangent functions).

Let $G$ be a regular graph. Let
$\mathsf{N}^{i},\mathsf{T}^{i}:[0,1]\to\mathbb{R}$ be functions of class
$C^{1}$, for $i=1,\ldots,N$. We say that the $\mathsf{T}^{i}$’s are _adapted_
to the $\mathsf{N}^{i}$’s whenever

$\mathsf{T}^{\ell}(|e^{\ell}_{m}-x|)=\chi(x)\mathscr{L}^{\ell}_{m}(\mathsf{N}^{i}(|e^{i}_{m}-x|),\mathsf{N}^{j}(|e^{j}_{m}-x|)),$
(3.2)

for $x\in[0,\tfrac{1}{2}]$ for any junction
$m=\pi(e^{i},i)=\pi(e^{j},j)=\pi(e^{k},k)$, and

$\mathsf{T}^{i}(x)=0$

for $x\in[\tfrac{1}{2},1]$ for any endpoint $\pi(1,i)$.

The next proposition introduce a canonical way of writing a one-parameter
family of triple junctions networks close in $H^{2}$ to a regular one
$\Gamma_{*}$ as graphs over $\Gamma_{*}$.
We shall apply the next result also to the particular case in which
$\Gamma_{t}\equiv\Gamma$ for any $t$.

######  Proposition 3.4 ([18, Proposition 3.4, Corollary 3.5]).

Let $\Gamma_{*}:G\to\mathbb{R}^{2}$ be a smooth regular network of a graph $G$
without endpoints. Then there exist $\varepsilon_{\Gamma_{*}}>0$ such that
whenever $\Gamma_{t}:G\to\mathbb{R}^{2}$ is a one-parameter family of triple
junctions networks of class $H^{2}$, differentiable with respect to $t$ for
$t\in[t_{0}-h,t_{0}+h]$ and $h>0$, such that
$(t,x)\mapsto\partial_{t}\gamma^{i}_{t}(x)$ is continuous for any $i$ and

$\displaystyle\sum_{i}\|\gamma^{i}_{*}-\gamma^{i}_{t}\|_{H^{2}(\mathrm{d}x)}\leq\varepsilon_{\Gamma_{*}},$

for any $i,t$, then there exist $h^{\prime}\in(0,h)$ and functions
$\mathsf{N}^{i}_{t},\mathsf{T}^{i}_{t}\in H^{2}(\mathrm{d}x)$ and
reparametrizations $\varphi^{i}_{t}:[0,1]\to[0,1]$ of class
$H^{2}(\mathrm{d}x)$, continuously differentiable with respect to $t$ for
$t\in[t_{0}-h^{\prime},t_{0}+h^{\prime}]$ such that

$\gamma^{i}_{t}\circ\varphi^{i}_{t}(x)=\gamma^{i}_{*}(x)+\mathsf{N}^{i}_{t}(x)\nu^{i}_{*}(x)+\mathsf{T}^{i}_{t}(x)\tau^{i}_{*}(x).$

Moreover, the $\mathsf{T}^{i}_{t}$’s are adapted to the $\mathsf{N}^{i}_{t}$’s
and

  * •

for any $\delta>0$ there is $\varepsilon\in(0,\varepsilon_{\Gamma_{*}})$ such
that if
$\sum_{i}\|\gamma^{i}_{*}-\gamma^{i}_{t}\|_{H^{2}(\mathrm{d}x)}\leq\varepsilon$
for all $t$, then

$\sum_{i}\|\mathsf{N}^{i}_{t}\|_{H^{2}(\mathrm{d}x)}+\|\varphi^{i}_{t}(x)-x\|_{H^{2}(\mathrm{d}x)}\leq\delta,$
(3.3)

for any $t\in[t_{0}-h^{\prime},t_{0}+h^{\prime}]$;

  * •

for any $\eta>0$ and $m\in\mathbb{N}$ there is
$\varepsilon_{\eta,m}\in(0,\varepsilon_{\Gamma_{*}})$ and $h_{\eta,m}\in(0,h)$
such that if
$\sum_{i}\|\gamma^{i}_{*}-\gamma^{i}_{t}\|_{C^{m+1}([0,1])}\leq\varepsilon$
for any $t$, then

$\sum_{i}\|\mathsf{N}^{i}_{t}\|_{H^{m}(\mathrm{d}x)}\leq\eta,$ (3.4)

for any $t\in[t_{0}-h_{\eta,m},t_{0}+h_{\eta,m}]$.

###  3.1 Łojasiewicz–Simon inequality for $\mathcal{G}$

In this section we prove the key Łojasiewicz–Simon inequality for the
functional $\mathcal{G}$. We need to compute first and second variations of
$\mathcal{G}$ taken with respect to variations on the normal functions
$\mathsf{N}^{i}$ (and, consequently, on the adapted tangent functions).

######  Proposition 3.5 (First Variation).

Let $\Gamma_{*}:G\to\mathbb{R}^{2}$ be a smooth regular network where $G$ is a
graph without endpoints. Then there is $\varepsilon_{\Gamma_{*}}>0$ such that
the following holds.

Let $\mathsf{N}^{i},X^{i}\in H^{2}$ with
$\|\mathsf{N}^{i}\|_{H^{2}}\leq\varepsilon_{\Gamma_{*}}$ such that

$\sum_{\ell\in
I_{m}}(-1)^{e^{\ell}_{m}}\mathsf{N}^{\ell}(e^{\ell}_{m})=\sum_{\ell\in
I_{m}}(-1)^{e^{\ell}_{m}}X^{\ell}(e^{\ell}_{m})=0\qquad\forall m\in J_{G}.$

Let $\Gamma^{\varepsilon}:G\to\mathbb{R}^{2}$ be the triple junctions network
defined by

$\gamma^{i,\varepsilon}(x)\vcentcolon=\gamma^{i}_{*}(x)+(\mathsf{N}^{i}(x)+\varepsilon
X^{i}(x))\nu^{i}_{*}(x)+\mathsf{T}^{i,\varepsilon}(x)\tau^{i}_{*}(x),$ (3.5)

for any $i$, for any $|\varepsilon|<\varepsilon_{0}$ and some
$\varepsilon_{0}>0$, where the $\mathsf{T}^{i,\varepsilon}$’s are adapted to
the $(\mathsf{N}^{i}+\varepsilon X^{i})$’s, for any
$|\varepsilon|<\varepsilon_{0}$.

Call $\Gamma$ the network given by the immersions
$\gamma^{i}\vcentcolon=\gamma^{i,0}$. Then

$\begin{split}\frac{\mathrm{d}}{\mathrm{d}\varepsilon}\mathcal{G}(\Gamma^{\varepsilon})\bigg{|}_{0}&=\sum_{m\in
J_{G}}\sum_{\ell\in
I_{m}}(-1)^{1+e^{\ell}_{m}}\left[\langle\tau^{\ell}(e^{\ell}_{m}),\nu^{\ell}_{*}(e^{\ell}_{m})\rangle+\sum_{j\in
I_{m}}h_{\ell
j}\langle\tau^{j}(e^{j}_{m}),\tau^{j}_{*}(e^{j}_{m})\rangle\right]e^{-|\gamma^{\ell}(e^{\ell}_{m})|^{2}/2}X^{\ell}(e^{\ell}_{m})+\\\
&\phantom{=}-\sum_{i}\int_{0}^{1}\bigg{(}\langle(\gamma^{i})^{\perp}+\boldsymbol{k}^{i},\nu^{i}_{*}\rangle
e^{-|\gamma^{i}|^{2}/2}|\partial_{x}\gamma^{i}|+\\\
&\qquad\qquad\qquad+\sum_{j}f_{ij}\chi\langle(\gamma^{j})^{\perp}+\boldsymbol{k}^{j},\tau^{j}_{*}\rangle
e^{-|\gamma^{j}|^{2}/2}|\partial_{x}\gamma^{j}|+\\\
&\qquad\qquad\qquad+g_{ij}\chi(1-x)\langle(\gamma^{j})^{\perp}+\boldsymbol{k}^{j},\tau^{j}_{*}\rangle(1-x)e^{-|\gamma^{j}(1-x)|^{2}/2}|\partial_{x}\gamma^{j}|(1-x)\bigg{)}X^{i}\,\mathrm{d}x,\end{split}$
(3.6)

where $h_{\ell,j},f_{ij},g_{ij}\in\mathbb{R}$ only depend on the topology of
$G$.

If also $\Gamma$ is regular, then

$\begin{split}\frac{\mathrm{d}}{\mathrm{d}\varepsilon}\mathcal{G}(\Gamma^{\varepsilon})\bigg{|}_{0}&=-\sum_{i}\int_{0}^{1}\bigg{(}\langle(\gamma^{i})^{\perp}+\boldsymbol{k}^{i},\nu^{i}_{*}\rangle
e^{-|\gamma^{i}|^{2}/2}|\partial_{x}\gamma^{i}|+\\\
&\qquad\qquad\qquad+\sum_{j}f_{ij}\chi\langle(\gamma^{j})^{\perp}+\boldsymbol{k}^{j},\tau^{j}_{*}\rangle
e^{-|\gamma^{j}|^{2}/2}|\partial_{x}\gamma^{j}|+\\\
&\qquad\qquad\qquad+g_{ij}\chi(1-x)\langle(\gamma^{j})^{\perp}+\boldsymbol{k}^{j},\tau^{j}_{*}\rangle(1-x)e^{-|\gamma^{j}(1-x)|^{2}/2}|\partial_{x}\gamma^{j}|(1-x)\bigg{)}X^{i}\,\mathrm{d}x.\end{split}$
(3.7)

###### Proof.

One gets (3.6) and (3.7) arguing analogously as in [18, Proposition 3.7]
directly computing the first variation of the functional $\mathcal{G}$, which
is just a length functional weighted by the function
$e^{-\frac{|\gamma^{i}(x)|^{2}}{2}}$. ∎

######  Proposition 3.6 (Second Variation).

Let $\Gamma_{*}:G\to\mathbb{R}^{2}$ be a smooth regular network where $G$ is a
graph without endpoints. Let $X^{i}\in H^{2}$ such that

$\sum_{\ell\in I_{m}}(-1)^{e^{\ell}_{m}}X^{\ell}(e^{\ell}_{m})=0\qquad\forall
m\in J_{G}.$

Let $\Gamma^{\varepsilon}:G\to\mathbb{R}^{2}$ be the triple junctions network
defined by

$\gamma^{i,\varepsilon}(x)\vcentcolon=\gamma^{i}_{*}(x)+\varepsilon
X^{i}(x)\nu^{i}_{*}(x)+\mathsf{T}^{i,\varepsilon}(x)\tau^{i}_{*}(x),$ (3.8)

for any $i$, for any $|\varepsilon|<\varepsilon_{0}$ and some
$\varepsilon_{0}>0$, where the $\mathsf{T}^{i,\varepsilon}$’s are adapted to
the $(\varepsilon X^{i})$’s, for any $|\varepsilon|<\varepsilon_{0}$

Then

$\begin{split}\frac{\mathrm{d}^{2}}{\mathrm{d}\varepsilon^{2}}\mathcal{G}(\Gamma^{\varepsilon})\bigg{|}_{0}&=\sum_{m\in
J_{G}}\sum_{\ell\in
I_{m}}(-1)^{1+e^{\ell}_{m}}\left(\partial_{s}X^{\ell}(e^{\ell}_{m})+\sum_{l\in
I_{m}}\omega^{l}_{m}(X^{l}(e^{l}_{m}))\right)X^{\ell}(e^{\ell}_{m})e^{-|\gamma^{\ell}_{*}(e^{\ell}_{m})|^{2}/2}+\\\
&\phantom{=}-\sum_{i}\int_{0}^{1}\bigg{(}\partial^{2}_{s}X^{i}+X^{i}+\sum_{j}\Omega_{ij}(\overline{X})\bigg{)}X^{i}e^{-|\gamma^{i}_{*}|^{2}/2}\,\mathrm{d}s_{\gamma^{i}_{*}},\end{split}$
(3.9)

where $\partial_{s}X^{n}=|\partial_{x}\gamma^{n}_{*}|^{-1}\partial_{x}X^{n}$
for any $n$, $\overline{X}\vcentcolon=(X^{1},\ldots,X^{N})$,
$\omega^{l}_{m}:\mathbb{R}\to\mathbb{R}$ is linear, and
$\Omega_{ij}:[H^{2}(0,1)]^{N}\to L^{2}(0,1)$ is a compact linear operator.

###### Proof.

Let us assume first that there is a junction $m$ such that the functions
$X^{\ell}$ appearing in (3.8) all vanish except for $\ell\in I_{m}$. Moreover,
for $\ell\in I_{m}$, assume that $X^{\ell}$ has compact support in
$[0,\tfrac{5}{8})$.

Let us denote $m=\pi(e^{i},i)=\pi(e^{j},j)=\pi(e^{k},k)$, where $i<j<k$.

Differentiating we get

$\begin{split}\frac{\mathrm{d}}{\mathrm{d}\varepsilon}\mathcal{G}(\Gamma^{\varepsilon})=\sum_{\ell\in
I_{m}}\int_{0}^{1}\left(-\langle\gamma^{\ell,\varepsilon},\partial_{\varepsilon}\gamma^{\ell,\varepsilon}\rangle+\frac{1}{|\partial_{x}\gamma^{\ell,\varepsilon}|^{2}}\langle\partial_{x}\gamma^{\ell,\varepsilon},\partial_{x}\partial_{\varepsilon}\gamma^{\ell,\varepsilon}\rangle\right)e^{-|\gamma^{\ell,\varepsilon}|^{2}/2}|\partial_{x}\gamma^{\ell,\varepsilon}|\,\mathrm{d}x.\end{split}$
(3.10)

Then setting

$Y^{\ell}\vcentcolon=\partial_{\varepsilon}\gamma^{\ell,\varepsilon}=X^{\ell}\nu^{\ell}_{*}+\chi\mathscr{L}^{\ell}_{m}(X^{i},X^{j})\tau^{\ell}_{*},$
(3.11)

we obtain

$\begin{split}\frac{\mathrm{d}^{2}}{\mathrm{d}\varepsilon^{2}}\mathcal{G}(\Gamma^{\varepsilon})\bigg{|}_{0}&=\sum_{\ell\in
I_{m}}\int_{0}^{1}\left(-|Y^{\ell}|^{2}-\frac{2}{|\partial_{x}\gamma^{\ell}_{*}|^{4}}\langle\partial_{x}\gamma^{\ell}_{*},\partial_{x}Y^{\ell}\rangle^{2}+\frac{1}{|\partial_{x}\gamma^{\ell}_{*}|^{2}}|\partial_{x}Y^{\ell}|^{2}\right)e^{-|\gamma^{\ell}_{*}|^{2}/2}|\partial_{x}\gamma^{\ell}_{*}|+\\\
&\qquad\qquad+\left(-\langle\gamma^{\ell}_{*},Y^{\ell}\rangle+\frac{1}{|\partial_{x}\gamma^{\ell}_{*}|^{2}}\langle\partial_{x}\gamma^{\ell}_{*},\partial_{x}Y^{\ell}\rangle\right)^{2}e^{-|\gamma^{\ell}_{*}|^{2}/2}|\partial_{x}\gamma^{\ell}_{*}|\,\mathrm{d}x\\\
&=\sum_{\ell\in
I_{m}}\int_{0}^{1}\left(-|Y^{\ell}|^{2}-2\langle\tau^{\ell}_{*},\partial_{s}Y^{\ell}\rangle^{2}+|\partial_{s}Y^{\ell}|^{2}\right)e^{-|\gamma^{\ell}_{*}|^{2}/2}+\\\
&\qquad\qquad+\left(-\langle\gamma^{\ell}_{*},Y^{\ell}\rangle+\langle\tau^{\ell}_{*},\partial_{s}Y^{\ell}\rangle\right)^{2}e^{-|\gamma^{\ell}_{*}|^{2}/2}\,\mathrm{d}s\\\
&=\sum_{\ell\in
I_{m}}(-1)^{e^{\ell}_{m}+1}\langle\partial_{s}Y^{\ell}(e^{\ell}_{m}),Y^{\ell}(e^{\ell}_{m})\rangle
e^{-|\gamma^{\ell}_{*}(e^{\ell}_{m})|^{2}/2}+\\\ &\qquad+\sum_{\ell\in
I_{m}}\int_{0}^{1}\left(-\langle\partial^{2}_{s}Y^{\ell},Y^{\ell}\rangle+\langle\partial_{s}Y^{\ell},Y^{\ell}\rangle\langle\gamma^{\ell}_{*},\tau^{\ell}_{*}\rangle-|Y^{\ell}|^{2}-2\langle\tau^{\ell}_{*},\partial_{s}Y^{\ell}\rangle^{2}\right)e^{-|\gamma^{\ell}_{*}|^{2}/2}+\\\
&\qquad\qquad+\left(-\langle\gamma^{\ell}_{*},Y^{\ell}\rangle+\langle\tau^{\ell}_{*},\partial_{s}Y^{\ell}\rangle\right)^{2}e^{-|\gamma^{\ell}_{*}|^{2}/2}\,\mathrm{d}s\\\
&=\sum_{\ell\in
I_{m}}(-1)^{e^{\ell}_{m}+1}\langle\partial_{s}Y^{\ell}(e^{\ell}_{m}),Y^{\ell}(e^{\ell}_{m})\rangle
e^{-|\gamma^{\ell}_{*}(e^{\ell}_{m})|^{2}/2}+\\\ &\qquad+\sum_{\ell\in
I_{m}}\int_{0}^{1}\bigg{(}-\langle\partial^{2}_{s}Y^{\ell},Y^{\ell}\rangle-|Y^{\ell}|^{2}-\langle\tau^{\ell}_{*},\partial_{s}Y^{\ell}\rangle^{2}+\\\
&\qquad\qquad+\langle\partial_{s}Y^{\ell},Y^{\ell}\rangle\langle\gamma^{\ell}_{*},\tau^{\ell}_{*}\rangle+\langle\gamma^{\ell}_{*},Y^{\ell}\rangle^{2}-2\langle\gamma^{\ell}_{*},Y^{\ell}\rangle\langle\tau^{\ell}_{*},\partial_{s}Y^{\ell}\rangle\bigg{)}e^{-|\gamma^{\ell}_{*}|^{2}/2}\,\mathrm{d}s.\end{split}$
(3.12)

We integrate by parts

$\begin{split}\sum_{\ell\in
I_{m}}&\int_{0}^{1}-\langle\tau^{\ell}_{*},\partial_{s}Y^{\ell}\rangle^{2}e^{-|\gamma^{\ell}_{*}|^{2}/2}\,\mathrm{d}s=\\\
&=\sum_{\ell\in
I_{m}}(-1)^{e^{\ell}_{m}}\langle\tau^{\ell}_{*},\partial_{s}Y^{\ell}\rangle\langle\tau^{\ell}_{*},Y^{\ell}\rangle
e^{-|\gamma^{\ell}_{*}|^{2}/2}\bigg{|}_{e^{\ell}_{m}}+\\\
&\qquad+\sum_{\ell\in
I_{m}}\int_{0}^{1}\bigg{(}\langle\tau^{\ell}_{*},\partial_{s}^{2}Y^{\ell}\rangle\langle\tau^{\ell}_{*},Y^{\ell}\rangle+\langle\boldsymbol{k}^{\ell}_{*},\partial_{s}Y^{\ell}\rangle\langle\tau^{\ell}_{*},Y^{\ell}\rangle+\langle\tau^{\ell}_{*},\partial_{s}Y^{\ell}\rangle\langle\boldsymbol{k}^{\ell}_{*},Y^{\ell}\rangle+\\\
&\qquad\qquad-\langle\tau^{\ell}_{*},\partial_{s}Y^{\ell}\rangle\langle\tau^{\ell}_{*},Y^{\ell}\rangle\langle\gamma^{\ell}_{*},\tau^{\ell}_{*}\rangle\bigg{)}e^{-|\gamma^{\ell}_{*}|^{2}/2}\,\mathrm{d}s.\end{split}$

Inserting into (3.12) we get

$\begin{split}\frac{\mathrm{d}^{2}}{\mathrm{d}\varepsilon^{2}}\mathcal{G}(\Gamma^{\varepsilon})\bigg{|}_{0}&=\sum_{\ell\in
I_{m}}(-1)^{e^{\ell}_{m}+1}\big{(}\langle\partial_{s}Y^{\ell}(e^{\ell}_{m}),Y^{\ell}(e^{\ell}_{m})\rangle-\langle\tau^{\ell}_{*}(e^{\ell}_{m}),\partial_{s}Y^{\ell}(e^{\ell}_{m})\rangle\langle\tau^{\ell}_{*}(e^{\ell}_{m}),Y^{\ell}(e^{\ell}_{m})\rangle\big{)}e^{-|\gamma^{\ell}_{*}(e^{\ell}_{m})|^{2}/2}+\\\
&\qquad+\sum_{\ell\in
I_{m}}\int_{0}^{1}\big{\langle}-\partial_{s}^{2}Y^{\ell}-\langle\partial_{s}^{2}Y^{\ell},\tau^{\ell}_{*}\rangle\tau^{\ell}_{*}-Y^{\ell}+\Omega^{\ell}(Y^{\ell}),Y^{\ell}\big{\rangle}e^{-|\gamma^{\ell}_{*}|^{2}/2}\,\mathrm{d}s,\end{split}$
(3.13)

where $\Omega^{\ell}:H^{2}(0,1)\to L^{2}(0,1)$ is a compact operator.
Inserting the definition of $Y^{\ell}$, we see that the second order term with
respect to $Y^{\ell}$ in (3.13) becomes

$\begin{split}-\partial_{s}^{2}Y^{\ell}-\langle\partial_{s}^{2}Y^{\ell},\tau^{\ell}_{*}\rangle\tau^{\ell}_{*}&=\partial_{s}^{2}X^{\ell}\nu^{\ell}_{*}+\partial^{2}_{s}(\chi\mathscr{L}^{\ell}_{m}(X^{i},X^{j}))\tau^{\ell}_{*}-\partial^{2}_{s}(\chi\mathscr{L}^{\ell}_{m}(X^{i},X^{j}))\tau^{\ell}_{*}+\sum_{j\in
I_{m}}\widetilde{\Omega}_{\ell j}(X^{j})\\\
&=\partial_{s}^{2}X^{\ell}\nu^{\ell}_{*}+\sum_{j\in
I_{m}}\widetilde{\Omega}_{\ell j}(X^{j}),\end{split}$

where $\widetilde{\Omega}_{\ell j}:H^{2}(0,1)\to[L^{2}(0,1)]^{2}$ is a compact
operator.
Moreover the boundary terms in (3.13) are

$\begin{split}\langle(\partial_{s}Y^{\ell})^{\perp},Y^{\ell}\rangle\big{|}_{e^{\ell}_{m}}&=\langle(\partial_{s}Y^{\ell})^{\perp},\nu^{\ell}_{*}\rangle
X^{\ell}\big{|}_{e^{\ell}_{m}}\\\
&=\partial_{s}X^{\ell}(e^{\ell}_{m})\,X^{\ell}(e^{\ell}_{m})+\mathscr{L}^{\ell}_{m}(X^{i}(e^{i}_{m}),X^{j}(e^{j}_{m}))\langle\boldsymbol{k}^{\ell}_{*}(e^{\ell}_{m}),\nu^{\ell}_{*}(e^{\ell}_{m})\rangle
X^{\ell}(e^{\ell}_{m}).\end{split}$

Hence (3.13) eventually implies (3.9). ∎

We now introduce the linear functional analytic setting needed for the proof
of the Łojasiewicz–Simon inequality.
For a fixed shrinker $\Gamma_{*}:G\to\mathbb{R}^{2}$, where $G$ is a graph
without endpoints, we denote by $M\vcentcolon=\sharp J_{G}$ the cardinality of
junctions, and we define the space

$\begin{split}V\vcentcolon=\bigg{\\{}\overline{\mathsf{N}}\vcentcolon=(\mathsf{N}^{1},\ldots,\mathsf{N}^{N})\in[H^{2}(0,1)]^{N}\
:\ &\sum_{\ell\in
I_{m}}(-1)^{e^{\ell}_{m}}\mathsf{N}^{\ell}(e^{\ell}_{m})=0\,\,\forall\,m\in
J_{G}\bigg{\\}},\end{split}$ (3.14)

endowed with the complete scalar product

$\langle\overline{\mathsf{N}},\overline{X}\rangle_{V}\vcentcolon=\sum_{i}\int_{0}^{1}\big{(}\mathsf{N}^{i}X^{i}+\partial_{s}^{2}\mathsf{N}^{i}\partial_{s}^{2}X^{i}\big{)}e^{-|\gamma^{i}_{*}|^{2}/2}\,\mathrm{d}s^{i}_{*},$

where $\mathrm{d}s^{i}_{*}\vcentcolon=\mathrm{d}s_{\gamma^{i}_{*}}$. We
further define the $2$-dimensional spaces

$W_{m}\vcentcolon=\left\\{(v^{\ell}_{m})_{\ell\in I_{m}}\in\mathbb{R}^{3}\ :\
\sum_{\ell\in I_{m}}(-1)^{e^{\ell}_{m}}v^{\ell}_{m}=0\right\\},$

for $m=1,\ldots,M$, where $W_{m}$ is endowed with the scalar product

$\langle(v^{\ell}_{m}),(w^{\ell}_{m})\rangle_{W_{m}}\vcentcolon=\sum_{\ell\in
I_{m}}v^{\ell}_{m}w^{\ell}_{m}e^{-|\gamma^{\ell}_{*}(e^{\ell}_{m})|^{2}/2}.$

Finally, let $Z$ be the product space

$Z\vcentcolon=\Pi_{m=1}^{M}W_{m}\times\Pi_{i=1}^{N}L^{2}(e^{-|\gamma^{i}_{*}|^{2}/2}\,\mathrm{d}s^{i}_{*})\eqqcolon\Pi_{m=1}^{M}W_{m}\times\Pi_{i=1}^{N}L^{2}(\mathrm{d}\mu^{i}_{*}).$
(3.15)

Observe that ${\rm j}:V\hookrightarrow Z$ compactly with the natural injection

$\overline{\mathsf{N}}\quad\overset{\rm
j}{\mapsto}\quad\left((\mathsf{N}^{\ell}(e^{\ell}_{m})),\overline{\mathsf{N}}\right)$
(3.16)

For $r_{\Gamma_{*}}>0$ small enough, we define the energy $\boldsymbol{{\rm
G}}:B_{r_{\Gamma_{*}}}(0)\subset V\to[0,+\infty)$ by

$\boldsymbol{{\rm
G}}(\overline{\mathsf{N}})\vcentcolon=\sum_{i}\mathcal{G}\left(\gamma^{i}_{*}+\mathsf{N}^{i}\nu^{i}_{*}+\mathsf{T}^{i}\tau^{i}_{*}\right),$
(3.17)

where the $\mathsf{T}^{i}$’s are adapted to the $\mathsf{N}^{i}$’s (see 3.3).
Recalling Lemma 3.1, the immersions
$\gamma^{i}_{*}+\mathsf{N}^{i}\nu^{i}_{*}+\mathsf{T}^{i}\tau^{i}_{*}$ define a
triple junctions network.

In the next lemma we check the validity of sufficient conditions implying the
desired inequality.

######  Lemma 3.7.

Let $\Gamma_{*}:G\to\mathbb{R}^{2}$ be a shrinker, where $G$ is a graph
without endpoints. Let $V,Z,\boldsymbol{\rm G}$ be as above, and identify
$Z^{\star}$ with ${\rm j}^{\star}(Z^{\star})\subset V^{\star}$, for ${\rm j}$
as in (3.16).

Then the following hold.

  1. 1.

The functional $\boldsymbol{{\rm G}}:B_{r_{\Gamma_{*}}}(0)\subset
V\to[0,+\infty)$ is analytic.

  2. 2.

The first variation operator $\delta\boldsymbol{\rm G}:V\to Z^{\star}$ is
$Z^{\star}$-valued by setting

$\begin{split}\delta&\boldsymbol{\rm
G}(\overline{\mathsf{N}})[((v^{\ell}_{m}),\overline{X})]\\\ &=\sum_{m\in
J_{G}}\left\langle\bigg{(}(-1)^{1+e^{\ell}_{m}}\bigg{[}\langle\tau^{\ell}(e^{\ell}_{m}),\nu^{\ell}_{*}(e^{\ell}_{m})\rangle+\sum_{j\in
I_{m}}h_{\ell
j}\langle\tau^{j}(e^{j}_{m}),\tau^{j}_{*}(e^{j}_{m})\rangle\bigg{]}e^{-|\gamma^{\ell}(e^{\ell}_{m})|^{2}/2}e^{|\gamma^{\ell}_{*}(e^{\ell}_{m})|^{2}/2}\bigg{)},(v^{\ell}_{m})\right\rangle_{W_{m}}\\\
&\quad-\sum_{i}\bigg{\langle}\langle(\gamma^{i})^{\perp}+\boldsymbol{k}^{i},\nu^{i}_{*}\rangle
e^{-|\gamma^{i}|^{2}/2}|\partial_{x}\gamma^{i}|e^{|\gamma^{i}_{*}|^{2}/2}|\partial_{x}\gamma^{i}_{*}|^{-1}+\\\
&\qquad\qquad+\sum_{j}f_{ij}\chi\langle(\gamma^{j})^{\perp}+\boldsymbol{k}^{j},\tau^{j}_{*}\rangle
e^{-|\gamma^{j}|^{2}/2}|\partial_{x}\gamma^{j}|e^{|\gamma^{i}_{*}|^{2}/2}|\partial_{x}\gamma^{i}_{*}|^{-1}+\\\
&\qquad\qquad+g_{ij}\chi(1-x)\langle(\gamma^{j})^{\perp}+\boldsymbol{k}^{j},\tau^{j}_{*}\rangle(1-x)e^{-|\gamma^{j}(1-x)|^{2}/2}|\partial_{x}\gamma^{j}|(1-x)e^{|\gamma^{i}_{*}|^{2}/2}|\partial_{x}\gamma^{i}_{*}|^{-1},X^{i}\bigg{\rangle}_{L^{2}(\mathrm{d}\mu^{i}_{*})}\end{split}$
(3.18)

where $f_{ij},g_{ij},h_{\ell j}\in\mathbb{R}$ depend on the topology of $G$,
and $\tau^{i},\boldsymbol{k}^{i}$ are referred to the immersions
$\gamma^{i}\vcentcolon=\gamma^{i}_{*}+\mathsf{N}^{i}\nu^{i}_{*}+\mathsf{T}^{i}\tau^{i}_{*}$,
with $\mathsf{T}^{i}$ adapted to $\mathsf{N}^{i}$.
Moreover $\delta\boldsymbol{\rm G}:V\to Z^{\star}$ is analytic.

If also the network defined by the immersions $\gamma^{i}$ is regular, then

$\begin{split}\delta&\boldsymbol{\rm
G}(\overline{\mathsf{N}})[((v^{\ell}_{m}),\overline{X})]\\\
&=-\sum_{i}\bigg{\langle}\langle(\gamma^{i})^{\perp}+\boldsymbol{k}^{i},\nu^{i}_{*}\rangle
e^{-|\gamma^{i}|^{2}/2}|\partial_{x}\gamma^{i}|e^{|\gamma^{i}_{*}|^{2}/2}|\partial_{x}\gamma^{i}_{*}|^{-1}+\\\
&\qquad\qquad+\sum_{j}f_{ij}\chi\langle(\gamma^{j})^{\perp}+\boldsymbol{k}^{j},\tau^{j}_{*}\rangle
e^{-|\gamma^{j}|^{2}/2}|\partial_{x}\gamma^{j}|e^{|\gamma^{i}_{*}|^{2}/2}|\partial_{x}\gamma^{i}_{*}|^{-1}+\\\
&\qquad\qquad+g_{ij}\chi(1-x)\langle(\gamma^{j})^{\perp}+\boldsymbol{k}^{j},\tau^{j}_{*}\rangle(1-x)e^{-|\gamma^{j}(1-x)|^{2}/2}|\partial_{x}\gamma^{j}|(1-x)e^{|\gamma^{i}_{*}|^{2}/2}|\partial_{x}\gamma^{i}_{*}|^{-1},X^{i}\bigg{\rangle}_{L^{2}(\mathrm{d}\mu^{i}_{*})}.\end{split}$
(3.19)

  3. 3.

The second variation $\delta^{2}\boldsymbol{\rm G}_{0}:V\to Z^{\star}$ at $0$
is $Z^{\star}$-valued by setting

$\begin{split}\delta^{2}\boldsymbol{\rm
G}_{0}(\overline{X})[((v^{\ell}_{m}),\overline{Z})]&=\sum_{m\in
J_{G}}\left\langle\bigg{(}(-1)^{1+e^{\ell}_{m}}\bigg{(}\partial_{s}X^{\ell}(e^{\ell}_{m})+\sum_{l\in
I_{m}}\omega^{l}_{m}(X^{l}(e^{l}_{m}))\bigg{)}\bigg{)},(v^{\ell}_{m})\right\rangle_{W_{m}}+\\\
&\phantom{=}-\sum_{i}\left\langle\partial^{2}_{s}X^{i}+X^{i}+\sum_{j}\Omega_{ij}(\overline{X}),Z^{i}\right\rangle_{L^{2}(\mathrm{d}\mu^{i}_{*})},\end{split}$
(3.20)

where $\partial_{s}X^{n}=|\partial_{x}\gamma^{n}_{*}|^{-1}\partial_{x}X^{n}$
for any $n$, $\overline{X}\vcentcolon=(X^{1},\ldots,X^{N})$,
$\omega^{l}_{m}:\mathbb{R}\to\mathbb{R}$ is linear, and $\Omega_{ij}:V\to
L^{2}(\mathrm{d}\mu^{i}_{*})$ is a compact linear operator.
Moreover $\delta^{2}\boldsymbol{\rm G}_{0}:V\to Z^{\star}$ is a Fredholm
operator of index zero.

###### Proof.

The proof follows by adapting the arguments in [18, Section 3.3].

  1. 1.

Analyticity of $\boldsymbol{\rm G}$ easily follows as in [18, Lemma 3.11].

  2. 2.

The expression for the first variation $\delta\boldsymbol{\rm G}$ and the fact
that it is $Z^{\star}$-valued follow from 3.5, while its analyticity follows
as in [18, Lemma 3.11].

  3. 3.

The expression for the second variation $\delta\boldsymbol{\rm G}_{0}$ at zero
and the fact that it is $Z^{\star}$-valued follow by polarizing the formula in
3.6.
Arguing as in [18, Lemma 3.12] one easily checks that the linear and
continuous operator $\mathscr{F}:V\to Z^{\star}$ given by

$\mathscr{F}(\overline{X})\vcentcolon={\rm
I}\left(\bigg{(}(-1)^{1+e^{\ell}_{m}}\bigg{(}\partial_{s}X^{\ell}(e^{\ell}_{m})-\frac{1}{3}\sum_{\ell\in
I_{m}}\partial_{s}X^{\ell}(e^{\ell}_{m})\bigg{)}\bigg{)},\bigg{(}-\partial^{2}_{s}X^{i}+X^{i}\bigg{)}\right),$

where ${\rm I}:Z\to Z^{\star}$ is the natural isometry, is invertible, and
thus it is Fredholm of index $0$. Recalling that Fredholmness is stable with
respect to addition of compact operators, since

$V\ni\overline{X}\quad\overset{\mathscr{F}_{1}}{\mapsto}\quad{\rm
I}\left(\bigg{(}(-1)^{e^{\ell}_{m}}\frac{1}{3}\sum_{\ell\in
I_{m}}\partial_{s}X^{\ell}(e^{\ell}_{m})\bigg{)},0\right)\in Z^{\star},$

and

$V\ni\overline{X}\quad\overset{\mathscr{F}_{2}}{\mapsto}\quad{\rm
I}\left(\bigg{(}(-1)^{1+e^{\ell}_{m}}\sum_{l\in
I_{m}}\omega^{l}_{m}(X^{l}(e^{l}_{m}))\bigg{)},\bigg{(}-2X^{i}+\sum_{j}\Omega_{ij}(\overline{X})\bigg{)}\right)\in
Z^{\star},$

are compact, then $\delta^{2}\boldsymbol{\rm
G}_{0}=\mathscr{F}-\mathscr{F}_{1}+\mathscr{F}_{2}$ is Fredholm of index zero.

∎

As a consequence of the previous lemma, we deduce the desired
Łojasiewicz–Simon inequality for the functional $\mathcal{G}$.

######  Corollary 3.8 (Łojasiewicz–Simon inequality at compact shrinkers).

Let $\Gamma_{*}:G\to\mathbb{R}^{2}$ be a shrinker, where $G$ is a graph
without endpoints. Then there exist $C_{\rm LS},\sigma>0$ and
$\theta\in(0,\tfrac{1}{2}]$ such that the following holds.

If $\Gamma:G\to\mathbb{R}^{2}$ is a regular network of class $H^{2}$ such that

$\left(\sum_{i}\|\gamma^{i}_{*}-\gamma^{i}\|_{H^{2}(\mathrm{d}x)}^{2}\right)^{\frac{1}{2}}\leq\sigma$

then

$\left|\mathcal{G}(\Gamma)-\mathcal{G}(\Gamma_{*})\right|^{1-\theta}\leq
C_{\rm
LS}\left(\sum_{i}\int_{\gamma^{i}}|(\gamma^{i})^{\perp}+\boldsymbol{k}^{i}|^{2}e^{-|\gamma^{i}|^{2}/2}\right)^{\frac{1}{2}}.$
(3.21)

###### Proof.

The proof immediately follows combining 3.4 with $\Gamma_{t}=\Gamma$ for any
$t$ and Lemma 3.7, applying [19, Corollary 2.6] which yields the desired
Łojasiewicz–Simon inequality as a consequence of the properties proved in
Lemma 3.7. For further details we refer to the strategy explained in [18,
Corollary 3.15]. ∎

###  3.2 Convergence of the rescaled motion by curvature

In this section we prove our main result 1.1, which is, in turn, implied by
the following theorem stating that subconvergence of a rescaled motion by
curvature improves to full convergence.

######  Theorem 3.9.

Let $\widetilde{\Gamma}_{\mathfrak{t}}$ be a rescaled motion by curvature at
the origin $0$ and maximal time of existence $T$ of a motion by curvature
$\Gamma_{t}:G\to\mathbb{R}^{2}$, where $G$ is a graph without endpoints.
Suppose that there is a sequence $\mathfrak{t}_{n}\to+\infty$ such that
$\widetilde{\Gamma}_{\mathfrak{t}_{n}}$ converges to a regular network
$\Gamma_{*}:G\to\mathbb{R}^{2}$ in $C^{k}$ on $G$ for any $k\in\mathbb{N}$, up
to reparametrizations.

Then there exists the full limit
$\lim_{\mathfrak{t}\to+\infty}\widetilde{\Gamma}_{\mathfrak{t}}=\Gamma_{*}$ in
$C^{k}$ for any $k$, up to reparametrizations.

###### Proof.

We first state some interpolation inequalities. For any $k\in\mathbb{N}$ with
$k\geq 1$ there exist $\lambda_{k}>0,\zeta_{k}\in(0,1)$ such that

$\|u\|_{H^{k}(\mathrm{d}x)}\leq\lambda_{k}\|u\|_{L^{2}(\mathrm{d}x)}^{\zeta_{k}}\|u\|_{H^{k+1}(\mathrm{d}x)}^{1-\zeta_{k}},$
(3.22)

for any $u\in H^{k+1}\left((0,1);\mathbb{R}^{N}\right)$. We shall drop the
subscript $k$ when $k=2$.

Next we observe that for any $k\in\mathbb{N}$ with $k\geq 3$ and $r>0$ there
is $C_{k}=C_{k}(r,\Gamma_{*})>0$ such that whenever
$\widehat{\Gamma}_{\mathfrak{t}}:G\to\mathbb{R}^{2}$ is a rescaled motion by
curvature with parametrizations
$\widehat{\gamma}^{i}_{\mathfrak{t}}=\gamma^{i}_{*}+\mathsf{N}^{i}_{\mathfrak{t}}\nu^{i}_{*}+\mathsf{T}^{i}_{\mathfrak{t}}\tau^{i}_{*}$,
i.e., there holds
$(\partial_{\mathfrak{t}}\widehat{\gamma}^{i}_{\mathfrak{t}})^{\perp}=(\widehat{\gamma}^{i}_{\mathfrak{t}})^{\perp}+\widehat{\boldsymbol{k}}^{i}_{\mathfrak{t}}$,
for $\overline{\mathsf{N}}_{\mathfrak{t}}\in B_{r}(0)\subset V$ in the
notation of (3.17), where the $\mathsf{T}^{i}_{\mathfrak{t}}$’s are adapted,
for $\mathfrak{t}\in[\tau_{1},\tau_{2}]$ with $\tau_{1}>1$, then

$\|\overline{\mathsf{N}}_{\mathfrak{t}}\|_{H^{k}(\mathrm{d}x)}\leq C_{k},$
(3.23)

for $\mathfrak{t}\in[\tau_{1},\tau_{2}]$.

Indeed, by 2.6 the motion $\widehat{\Gamma}_{\mathfrak{t}}$ corresponds to a
parabolically rescaled motion by curvature $\Gamma^{\mu}_{t}$ such that
$\Gamma^{\mu}_{-\frac{1}{2}}=\widehat{\Gamma}_{\tau_{1}}$ via the identities

$\Gamma^{\mu}_{t}=\sqrt{-2t}\,\widehat{\Gamma}_{\log(\mu/\sqrt{-t})}\qquad\mu^{2}=\frac{e^{2\tau_{1}}}{2}.$

Let $\overline{t}\in(-1/2,0)$ be such that
$\tau_{2}=\log\left(\frac{\mu}{\sqrt{-\overline{t}}}\right)$. Assume first
that $\overline{t}\leq-\tfrac{1}{4}$. Then $\Gamma^{\mu}_{t}$ is a motion by
curvature for $t\in[-1/2,\overline{t}]$ with curvature uniformly bounded in
$L^{2}(\mathrm{d}s)$ and with length of the edges uniformly bounded from below
away from zero. Recalling [15, Proposition 5.8] and since
$\sqrt{-2t}\in[1/\sqrt{2},1]$ for $t\in[-1/2,\overline{t}]$, it is readily
checked that this implies (3.23) (we refer the reader to the argument in the
proof of [18, Theorem 5.2]). In case $\overline{t}>-\tfrac{1}{4}$, then the
previous argument implies estimates (3.23) for
$\mathfrak{t}\in[\tau_{1},\log(\mu/\sqrt{1/4})]\subset[\tau_{1},\tau_{2}]$.
Hence the argument can be iterated with $\log(\mu/\sqrt{1/4})$ in place of
$\tau_{1}$. Iterating again and again finitely many times we get that (3.23)
holds on $[\tau_{1},\tau_{2}]$.

The proof continues adapting the strategy of [18, Theorem 5.2]. Without loss
of generality we can assume that
$\mathcal{G}(\widetilde{\Gamma}_{\mathfrak{t}})>\mathcal{G}(\Gamma_{*})$ for
any $\mathfrak{t}$, for otherwise the rescaled motion by curvature would be
stationary, and the thesis would follow trivially.

Let $\sigma,\theta,C_{\rm LS}$ be as in 3.8. For $r_{\Gamma_{*}}>0$ suitably
small, whenever
$\widehat{\gamma}^{i}\vcentcolon=\gamma^{i}_{*}+\mathsf{N}^{i}\nu^{i}_{*}+\mathsf{T}^{i}\tau^{i}_{*}$
is a smooth regular network, for $\overline{\mathsf{N}}\in
B_{r_{\Gamma_{*}}}(0)\subset V$ in the notation of (3.17), where the
$\mathsf{T}^{i}$’s are adapted, then

  1. 1)

there is $r_{\sigma}\in(0,r_{\Gamma_{*}})$ such that if $\overline{N}\in
B_{r_{\sigma}}(0)\subset V$, then
$\left(\sum_{i}\|\gamma^{i}_{*}-\widehat{\gamma}^{i}\|_{H^{2}(\mathrm{d}x)}^{2}\right)^{\frac{1}{2}}\leq\sigma$;

  2. 2)

there is $e_{*}>0$ such that $e^{-|\widehat{\gamma}^{i}|^{2}/2}\geq e_{*}^{2}$
for any $i$;

  3. 3)

there exists a constant $C_{G}>0$, depending only on the graph $G$ and
$\Gamma_{*}$, such that

$\begin{split}&\langle\widehat{\nu}^{i},\nu^{i}_{*}\rangle\geq\frac{3}{4},\qquad\qquad|\langle\widehat{\nu}^{i},\tau^{i}_{*}\rangle|<\frac{1}{C_{G}},\\\
&\sum_{m\in J_{G}}\sum_{\ell\in
I_{m}}\left|a^{\ell}(x)\langle\widehat{\nu}^{\ell},\nu^{\ell}_{*}\rangle+\langle\widehat{\nu}^{\ell},\tau^{\ell}_{*}\rangle\chi(x)\mathscr{L}^{\ell}_{m}(a^{i_{m}}(x),a^{j_{m}}(x))\right|^{2}\geq\frac{1}{C_{G}}\sum_{i}|a^{i}(x)|^{2},\end{split}$

where $\widehat{\nu}^{i}$ is the normal vector of $\widehat{\gamma}^{i}$, and
$i_{m}$ (resp. $j_{m}$) denotes the minimal (resp. intermediate) element of
$I_{m}$, for any continuous functions $a^{1},\ldots,a^{N}$;

  4. 4)

there exist $c_{1},c_{2}>0$ such that

$c_{1}\leq|\partial_{x}\widehat{\gamma}^{i}|^{-1}\leq c_{2},$

for any $i$.

For any $k\in\mathbb{N}$ with $k\geq 3$ let $C_{k}$ be given by (3.23) with
$r=r_{\sigma}$.
By assumptions and by 3.4 there exist times
$\mathfrak{t}_{2}>\mathfrak{t}_{1}>0$ and
$\mathsf{N}^{i}_{\mathfrak{t}},\mathsf{T}^{i}_{\mathfrak{t}},\varphi^{i}_{\mathfrak{t}}$
as in 3.4 such that

$(\mathcal{G}(\widetilde{\Gamma}_{\mathfrak{t}_{1}})-\mathcal{G}(\Gamma_{*}))^{\theta}<\frac{\theta
e_{*}}{(100\lambda C_{3})^{\frac{1}{\zeta}}C_{\rm
LS}\sqrt{C_{G}\,c_{2}}}r_{\sigma}^{\frac{1}{\zeta}}$ (3.24)
$\begin{split}\widetilde{\gamma}^{i}_{\mathfrak{t}}\circ\sigma^{i}\circ\varphi^{i}_{\mathfrak{t}}=\gamma^{i}_{*}+\mathsf{N}^{i}_{\mathfrak{t}}\nu^{i}_{*}+\mathsf{T}^{i}_{\mathfrak{t}}\tau^{i}_{*}\eqqcolon\widehat{\gamma}^{i}_{\mathfrak{t}},\qquad\overline{\mathsf{N}}_{\mathfrak{t}}\in
B_{\frac{r_{\sigma}}{2}}(0)\subset V,\end{split}$ (3.25)

for any $\mathfrak{t}\in[\mathfrak{t}_{1},\mathfrak{t}_{2}]$. Let
$S\in[\mathfrak{t}_{2},+\infty]$ be the supremum of times $s$ such that (3.25)
holds on $\mathfrak{t}\in[\mathfrak{t}_{1},s]$ for some
$\mathsf{N}^{i}_{\mathfrak{t}},\mathsf{T}^{i}_{\mathfrak{t}},\varphi^{i}_{\mathfrak{t}}$
as in 3.4.

Letting
$H(\mathfrak{t})\vcentcolon=(\mathcal{G}(\widetilde{\Gamma}_{\mathfrak{t}})-\mathcal{G}(\Gamma_{*}))^{\theta}>0$,
we can differentiate $H$ to get

$\begin{split}-\frac{\mathrm{d}}{\mathrm{d}\mathfrak{t}}H(\mathfrak{t})&=\theta
H^{\frac{\theta-1}{\theta}}(\mathfrak{t})\sum_{i}\int_{\widetilde{\gamma}^{i}_{\mathfrak{t}}}|(\widetilde{\gamma}^{i}_{\mathfrak{t}})^{\perp}+\widetilde{\boldsymbol{k}}^{i}_{\mathfrak{t}}|^{2}e^{-|\widetilde{\gamma}^{i}_{\mathfrak{t}}|^{2}/2}\\\
&=\theta
H^{\frac{\theta-1}{\theta}}(\mathfrak{t})\bigg{(}\sum_{i}\int_{\widehat{\gamma}^{i}_{\mathfrak{t}}}|(\widehat{\gamma}^{i}_{\mathfrak{t}})^{\perp}+\widehat{\boldsymbol{k}}^{i}_{\mathfrak{t}}|^{2}e^{-|\widehat{\gamma}^{i}_{\mathfrak{t}}|^{2}/2}\bigg{)}^{\frac{1}{2}}\left\|(\partial_{\mathfrak{t}}\widehat{\Gamma}_{\mathfrak{t}})^{\perp}\right\|_{L^{2}(\widehat{\mu}_{t})}\\\
&\overset{\ref{it:r1}}{\geq}\frac{\theta}{C_{\rm
LS}}\left\|(\partial_{\mathfrak{t}}\widehat{\Gamma}_{\mathfrak{t}})^{\perp}\right\|_{L^{2}(\widehat{\mu}_{t})},\end{split}$

for any $\mathfrak{t}\in[\mathfrak{t}_{1},S)$, where we denoted
$\left\|(\partial_{\mathfrak{t}}\widehat{\Gamma}_{\mathfrak{t}})^{\perp}\right\|_{L^{2}(\widehat{\mu}_{\mathfrak{t}})}^{2}\vcentcolon=\sum_{i}\int|(\partial_{t}\widehat{\gamma}^{i}_{\mathfrak{t}})^{\perp}|^{2}\,\mathrm{d}\widehat{\mu}^{i}_{\mathfrak{t}}$,
and
$\widehat{\mu}^{i}_{\mathfrak{t}}\vcentcolon=e^{-|\widehat{\gamma}^{i}_{\mathfrak{t}}|^{2}/2}|\partial_{x}\widehat{\gamma}^{i}_{\mathfrak{t}}|\,\mathrm{d}x$,
and we applied 3.8 thanks to 1). Exploiting (3.25), the above estimate becomes

$\begin{split}-\frac{\mathrm{d}}{\mathrm{d}\mathfrak{t}}H&\geq\frac{\theta}{C_{\rm
LS}}\left(\sum_{i}\int_{\widehat{\gamma}^{i}_{\mathfrak{t}}}|(\partial_{\mathfrak{t}}\widehat{\gamma}^{i}_{\mathfrak{t}})^{\perp}|^{2}e^{-|\widehat{\gamma}^{i}_{\mathfrak{t}}|^{2}/2}\right)^{\frac{1}{2}}\overset{\ref{it:r1bis}}{\geq}\frac{\theta
e_{*}}{C_{\rm
LS}}\left(\sum_{i}\int_{\widehat{\gamma}^{i}_{\mathfrak{t}}}|\partial_{\mathfrak{t}}\mathsf{N}^{i}_{\mathfrak{t}}\langle\nu^{i}_{\mathfrak{t}},\nu^{i}_{*}\rangle+\partial_{\mathfrak{t}}\mathsf{T}^{i}_{\mathfrak{t}}\langle\nu^{i}_{\mathfrak{t}},\tau^{i}_{*}\rangle|^{2}\right)^{\frac{1}{2}}\\\
&\geq\frac{\theta e_{*}}{C_{\rm LS}}\left(\frac{1}{2}\sum_{m\in
J_{G}}\sum_{\ell\in
I_{m}}\int_{\widehat{\gamma}^{\ell}_{\mathfrak{t}}}|\partial_{\mathfrak{t}}\mathsf{N}^{\ell}_{\mathfrak{t}}\langle\nu^{\ell}_{\mathfrak{t}},\nu^{\ell}_{*}\rangle+\partial_{\mathfrak{t}}\mathsf{T}^{\ell}_{\mathfrak{t}}\langle\nu^{\ell}_{\mathfrak{t}},\tau^{\ell}_{*}\rangle|^{2}\right)^{\frac{1}{2}}\\\
&\overset{\ref{it:r2}}{\geq}\frac{\theta e_{*}}{C_{\rm
LS}\sqrt{C_{G}}}\left(\sum_{i}\int_{\widehat{\gamma}^{i}_{\mathfrak{t}}}|\partial_{\mathfrak{t}}\mathsf{N}^{i}_{\mathfrak{t}}|^{2}\right)^{\frac{1}{2}}\\\
&\overset{\ref{it:r3}}{\geq}\frac{\theta e_{*}}{C_{\rm
LS}\sqrt{C_{G}\,c_{2}}}\left(\sum_{i}\int_{0}^{1}|\partial_{\mathfrak{t}}\mathsf{N}^{i}_{\mathfrak{t}}|^{2}\,\mathrm{d}x\right)^{\frac{1}{2}},\end{split}$

for any $\mathfrak{t}\in[\mathfrak{t}_{1},S)$. Hence

$\displaystyle\left\|\overline{\mathsf{N}}_{s_{2}}-\overline{\mathsf{N}}_{s_{1}}\right\|_{L^{2}(\mathrm{d}x)}$
$\displaystyle\leq\int_{s_{1}}^{s_{2}}\left\|\partial_{\mathfrak{t}}\overline{\mathsf{N}}_{\mathfrak{t}}\right\|_{L^{2}(\mathrm{d}x)}\,\mathrm{d}t\leq\frac{C_{\rm
LS}\sqrt{C_{G}\,c_{2}}}{\theta e_{*}}H(\mathfrak{t}_{1})$ (3.26)
$\displaystyle\overset{\eqref{eq:zzxxz}}{\leq}\frac{r_{\sigma}^{\frac{1}{\zeta}}}{(100\lambda
C_{3})^{\frac{1}{\zeta}}},$ (3.27)

for any $\mathfrak{t}_{1}\leq s_{1}<s_{2}<S$. By (3.22) with $k=2$, and by
(3.23) we obtain

$\left\|\overline{\mathsf{N}}_{s_{2}}-\overline{\mathsf{N}}_{s_{1}}\right\|_{H^{2}(\mathrm{d}x)}\leq\frac{r_{\sigma}}{50},$

for any $\mathfrak{t}_{1}\leq s_{1}<s_{2}<S$. Since
$\overline{\mathsf{N}}_{\mathfrak{t}_{1}}\in
B_{\frac{r_{\sigma}}{2}}(0)\subset V$, arguing by contradiction one gets that
$S=+\infty$. Since $H(\mathfrak{t})\to 0$ as $\mathfrak{t}\to+\infty$, then
(3.26) implies that $\overline{\mathsf{N}}_{\mathfrak{t}}$ is Cauchy in
$L^{2}(\mathrm{d}x)$ for $\mathfrak{t}\to+\infty$. Interpolating as before, we
deduce that there exists the full limit
$\lim_{\mathfrak{t}\to+\infty}\widetilde{\Gamma}_{\mathfrak{t}}=\Gamma_{*}$ in
$C^{k}$ for any $k$. ∎

###### Proof of 1.1.

The statement follows from 3.9 together with the equivalence recalled in 2.6.
∎

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