On the curve diffusion flow of closed plane curves

0.23 MB·English

by  Glen Wheeler

ON THE CURVE DIFFUSION FLOW OF CLOSED PLANE
CURVES
2
1 GLENWHEELER
0
2 Abstract. InthispaperweconsiderthesteepestdescentH−1-gradientflow

n ofthelengthfunctionalforimmersedplanecurves,knownasthecurvediffusion
a flow. Itisknownthatunderthisflowthereexistbothinitiallyimmersedcurves
J which develop at least one singularity in finite time and initially embedded

8 curveswhichself-intersectinfinitetime. Weprovethatundertheflowclosed
1 curves with initial data close to a round circle in the sense of normalised L2
oscillation of curvature exist for all time and converge exponentially fast to

] a round circle. This implies that for a sufficiently large ‘waiting time’ the
P evolving curves are strictly convex. We provide an optimal estimate for this
A waitingtime, whichgives aquantified feelingforthemagnitude to whichthe
maximum principle fails. We are also able to control the maximum of the
.
h multiplicity of the curve along the evolution. A corollary of this estimate is
t thatinitiallyembeddedcurvessatisfyingthehypothesesoftheglobalexistence
a
theorem remain embedded. Finally, as an application we obtain a rigidity
m
statement forclosedplanarcurveswithwindingnumberone.
[

1
v
5 1. Introduction
3
Supposeγ :S1 R2 isanimmersedclosedplanecurveofperiodP andconsider
7
→
3 the energy
. P
1 L(γ)= γ du,
u
0 | |
Z0
2 where γ = ∂ γ. We wish to deform γ towards a minimiser of L, and for this
u u
1 purpose we shall consider the steepest descent gradient flow of L in H−1. There
:
v are some advantages in choosing H−1 instead of L2. One is that for any initial
i curve the signed area is constant under the flow, which implies that if the signed
X
area of the initial curve is non-zero, then the flow is never asymptotic to a lower
r
a dimensional subset of R2.
The Euler-Lagrangeoperator of L in H−1 is
gradH−1L(γ)=kss,
wherek = γ ,ν isthecurvatureofγ, ν aunitnormalvectorfieldonγ,andsde-
ss
h i
notes arc-length. The curvediffusion flowis the one-parameterfamily ofimmersed
curves γ :S1×[0,T)→R2 with normal velocity equal to −gradH−1(L(γ)), that is
(CD) ∂⊥γ = k .
t − ss
Thecurvediffusionflowisadegeneratesystemofquasilinearfourthorderparabolic
partial differential equations, and as such it is not expected that a maximum or
comparisonprinciple holds. Indeed, GigaandIto[25]providedthe firstexample of
2000 Mathematics Subject Classification. 53C44and58J35.
FinancialsupportfromtheAlexander-von-HumboldtStiftungisgratefullyacknowledged.
1
2 GLENWHEELER
asimple,closed,strictlyembeddedplanarcurvewhichdevelopsaself-intersectionin
finite timeunder theflow. They alsogave[26]the firstexampleofasimple,closed,
strictlyconvexplanarcurvewhichbecomesnon-convexinfinitetime. Furthermore,
ElliotandMaier-Paapeshowed[15]thatthecurvediffusionflowmaydriveaninitial
graphto become non-graphicalin finite time. It was eventually shown by Blatt [8]
thatnon-preservationofconvexityandnon-preservationofembeddednessisabasic
property of a large class of general higher order hypersurface flows.
It is also known (see Polden [36] for the first example and Escher-Ito [16] for
many others) that the curve diffusion flow can from smooth immersed initial data
develop finite time curvature singularities. In contrast, our goal in this paper is to
demonstrateanewclassofinitialdata(generalising[14,Theorem6.1])whichgives
rise to an immortalsolutionconvergingexponentially fast to a simple roundcircle.
The curve diffusion flow has been considered for some time in the literature.
The first point to note is that for regular enough initial data γ : S1 R2 there
0
is a maximal T (0, ] and corresponding solution γ : S1 [0,T) →R2 which
∈ ∞ × →
satisfies (CD). Local existence, although technical and sometimes tricky, is by
now standard—inthis paper we state a version(Theorem2.1) which is a combina-
tion of Elliot-Garcke [14] and Dziuk-Kuwert-Scha¨tzle [11, Theorem 3.1], although
similar results appeared earlier, see [6, 9, 14, 39] for example. It is also quite stan-
dardregardless: asmentioned, the evolutionequation(CD) is adegeneratefourth-
orderquasilinearparabolicsystem,andlocalexistencecanbeobtainedforexample
through the method of semigroups (Angenent [4], Amann [1, 2, 3], Escher-Meyer-
Simonett [17, 19], and Lunardi [35] are good references), the Nash-Moser inverse
functiontheorem(see Hamilton[27,28], andGage-Hamilton[22])orthroughmore
classicalmethods suchascanbe foundin Polden[36]and Huisken-Polden[29] (see
also Sharples [37] and the books [12, 13, 21]). The local existence theorem we use
requires that the curvature of γ lies in L2. One should note that there are local
existence results which do not require any control of curvature, instead requiring
Lipschitz with small Lipschitz constant or slightly more regularity than C1 for the
initial data, see Koch-Lamm [30], Escher-Mucha [18], and Asai [5] for example.
The analysis we present here is direct and geometric in nature, and should be
compared with [7, 10, 11, 28, 31, 32, 33, 36, 40]. It rests on the observation that
the normalised oscillation of curvature
K γ(,t) =L γ(,t) (k k)2ds,
osc
· · −
Zγ
(cid:0) (cid:1) (cid:0) (cid:1)
wherek denotestheaverageofthecurvature,isinmanyrespectsanatural‘energy’
for the flow. The only stationary solutions of (CD) are lines and multiply covered
t
circles, for which K =0. Further, for arbitrary smooth initial data K dτ
osc 0 osc ≤
L4(γ(,0))/16π2 (see Lemma 3.2), that is, K L1 [0,T) .
· osc ∈ R
WeprovethatifK isinitiallysmallandtheisoperimetricratioI =L2/4πAis
osc (cid:0) (cid:1)
initiallyclosetoone,thentheyremainso. Thisisenoughtobegina‘bootstrapping’
style procedure, in which we use interpolation inequalities as in [11] to obtain
uniform bounds for all higher derivatives of curvature. These observations and
some extra arguments give the global existence result of this paper.
ON THE CURVE DIFFUSION FLOW OF CLOSED PLANE CURVES 3
Theorem 1.1. Suppose γ : S1 R2 is a regular smooth immersed closed curve
0
→
with A(γ )>0 and
0
(1) kds=2π.
Zγ0
There exists a constant K∗ >0 such that if
K∗
(2) K (γ )<K∗, and I(γ )<exp ,
osc 0 0 8π2
then the curve diffusion flow γ : S1 [0,T) R2 with γ(cid:16) as i(cid:17)nitial data exists for
0
× →
all time and converges exponentially fast to a round circle with radius A(0).
π
q
Remark 1.2. One advantage of our direct method is that we are able to easily
find an allowable choice for the constant K∗ above; in particular, one may select
2π+12π2 4π√3π√1+3π 1
K∗ = − .
3 ≃ 18
Remark 1.3. So long as A(γ ) = 0, one may always guarantee A(γ ) > 0 by
0 0
6
reversing the orientation of ν, since (CD) is invariant under change of orientation.
Remark 1.4. As can be seen from the proof of Proposition 3.7, the smallness
condition (2) could be weakened to
K (γ )+8π2log I(γ ) 2K∗ δ,
osc 0 0
≤ −
foranyδ >0. We donotexpectthis tobpe optimal,however. Atthistime, itisnot
known if there exists any smooth plane curve satisfying (1) which gives rise to a
curve diffusion flow with finite maximal existence time. Without at least one such
singular example, it is difficult to even conjecture on what an optimal form of (2)
may be.
It is clear that Theorem 1.1 implies k(,t) π , and so after a fixed time
· → A(0)
translation we have q
k(,t) √c>0
· ≥
foranyc (0, π )(cf. [31,Lemma5.5]and[41]fortheWillmoreflowandsurface
∈ A(0)
diffusion flow of surfaces respectively). In other words, after some finite time the
curvature becomes positive and remains so. This can be thought of as ‘eventual
positivity’,andisreminiscentofthesituationconsideredin[20,23,24]. There,using
very different techniques, eventual local positivity and other related qualitative
propertiesareobservedforbiharmonicparabolicequationsundercertainconditions.
To further quantify the size of the ‘waiting time’, we present the following.
Proposition 1.5. Suppose γ : S1 [0,T) R2 solves (CD) and satisfies the
× →
assumptions of Theorem 1.1. Then
L(γ ) 4 A(γ ) 2
0 0
t [0, ):k(,t)>0 .
L ∈ ∞ · 6 ≤ 2π − π
In the above, k(cid:8)(,t)> 0 means that (cid:9)there(cid:16)exists(cid:17)a p s(cid:16)uch tha(cid:17)t k(p,t) 0. This
· 6 ≤
estimate is optimal in the sense that the right hand side is zero for a simple circle.
ItisnotclearatallfromTheorem1.1ifinitiallyembeddedcurvesremainso,nor
even if we can control the maximum of the multiplicity (the number of times the
curve intersects itself in one point) of the evolving curve. We do havegood control
4 GLENWHEELER
of the oscillation of curvature however, and in the spirit of [34, Theorem 6] (see
also the monotonicity formula in [38] and appendix of [33]) present the following
theorem to address this issue.
Theorem 1.6. Suppose γ : S1 R2 is a smooth immersed curve with winding
→
number ω and let m denote the maximum number of times γ intersects itself in any
one point; that is
kds=2ωπ and m(γ)= sup γ−1(x).
Zγ x∈R2| |
Then
K (γ) 16m2 4ω2π2.
osc
≥ −
When combined with Proposition 3.7 we obtain the following.
Corollary 1.7. Any curve diffusion flow γ : S1 [0,T) R2 with initial data
γ :S1 R2 satisfying the assumptions of Theorem×1.1 wit→h
0
→
K∗ <64 4π2 24.5
− ≃
remains embedded for all time.
NotethatinparticulartheallowablechoiceforK∗ giveninRemark1.2issmaller
than 64 4π2.
−
Theorem1.1givesaone-parameterfamilyofsmoothdiffeomorphismsconnecting
the initial data γ with a round circle. This implies the following rigidity result.
0
Corollary 1.8. Let γ :S1 R2 be a regular closed immersed curve satisfying the
→
assumptions of Theorem 1.1. Then γ is diffeomorphic to a round circle.
This paper is organised as follows. In Section 2 we fix our notation, state the
local existence theorem, and prove some elementary Sobolev-Poincar´e-Wirtinger
inequalities. Section 3 contains estimates for the curvature in L2 and the isoperi-
metric ratio under various assumptions,which forms the bulk of the work involved
in proving Theorem 1.1. The theorem itself and Proposition1.5 are provedin Sec-
tion 4. We finish the paper by proving Theorem 1.6 and Corollary 1.7 in Section
5.
Acknowledgements
The author thanks his colleagues for several useful discussions, in particular
Hans-Christoph Grunau for reading an early version of this paper. The author
would also like to thank Ernst Kuwert for helpful discussions at the Mathema-
tisches Forschungsinstitut Oberwolfach (MFO). This work was completed under
the financial support of the Alexander von Humboldt Stiftung at the Otto-von-
Guericke-Universit¨atMagdeburg.
2. Preliminaries
Suppose γ : R R2 is a regular smooth immersed plane curve. We say that γ
isperiodicwithpe→riodP ifthereexistsavectorV R2 andapositiveP suchthat
for all m N ∈
∈
γ(u+P)=γ(u)+V, and ∂mγ(u+P)=∂mγ(u).
u u
ON THE CURVE DIFFUSION FLOW OF CLOSED PLANE CURVES 5
If V = 0 then γ is closed. In this case γ is an immersed circle, γ : S1 R2. The
→
length of γ is
P
L(γ)= γ du,
u
| |
Z0
and the signed enclosed area is
1 P
(3) A(γ)= γ,ν γ du,
u
−2 h i| |
Z0
where ν is a unit normal vector field on γ. Throughout the paper we keep γ
parametrised by arc-length s, where ds = γ du. Integrals over γ are to be inter-
u
| |
preted as integrals over the interval of periodicity.
Considerthe one-parameterfamilyofimmersedcurvesγ :S1 [0,T) R2 with
× →
normal velocity equal to −gradH−1(L(γ)), that is
(CD) ∂⊥γ = k .
t − ss
The followingtheoremisstandard. The uniquenessbelowisunderstoodmodulo
the natural group of invariances enjoyed by (CD): rotations, translations, changes
of orientation, and so on, as is customary for geometric flows.
Theorem 2.1 (Local existence). Suppose γ : R R2 is a periodic regular curve
0
→
parametrisedbyarc-lengthandofclass C1 W2,2 with k < . Then thereexists
2
a T (0, ] and a unique one-parameter f∩amily of immkekrsion∞s γ :R [0,T) R2
∈ ∞ × →
parametrised by arc-length such that
(i) γ(0, )=γ ;
0
·
(ii) ∂⊥γ = k ;
t − ss
(iii) γ(,t) is of class C∞ and periodic of period L(γ(,t)) for every t (0,T);
· · ∈
(iv) T is maximal.
Theorem 2.1 justifies the use of smooth calculations in the derivation of our
estimates. When we use the expression “γ : S1 [0,T) R2 solves (CD)” we are
× →
invoking Theorem 2.1 in the special case where the initial data is assumed to be
closed, but not necessarily embedded..
We will need the following elementary Sobolev-Poincar´e-Wirtingerinequalities.
Lemma 2.2. Suppose f :R R is absolutely continuous and periodic with period
P. Then if P fdx=0 we h→ave
0
R P P2 P
f2dx f 2dx,
≤ 4π2 | x|
Z0 Z0
with equality if and only if f(x)=asin(2xπ/P +b).
Proof. Expand f as a Fourier series and then use Parseval’s identity. (cid:3)
Corollary 2.3. Under the assumptions of Lemma 2.2,
P
f 2 f 2.
k k∞ ≤ 2πk xk2
Proof. As f has zero average, there exist p ,p such that f(p ) = f(p ) = 0 and
1 2 1 2
0 p <p <P. Thus, since f is absolutely continuous and periodic,
1 2
≤
x p2
f2(x)= f(u)f (u)du f(u)f (u)du.
x x
−
Zp1 Zx
6 GLENWHEELER
Therefore
p2 P
f2(x) f(u)f (u) du f(u)f (u) du.
x x
≤ | | ≤ | |
Zp1 Z0
Now Ho¨lder’s inequality and Lemma 2.2 above implies
P
f 2 f f f 2,
k k∞ ≤k k2k xk2 ≤ 2πk xk2
as required. (cid:3)
As most of our analysis is based on integral estimates, it is efficient to first
compute the derivative of an integral along the flow in general.
Lemma 2.4. Suppose γ :S1 [0,T) R2 solves (CD), and f :S1 [0,T) R is
× → × →
a periodic function with the same period as γ. Then
d
fds= f +fkk f (∂⊤γ)ds.
dt t ss− s t
Zγ Zγ
Proof. First note that τ = γ /γ = γ is a unit tangent vector field along γ. We
u u s
| |
compute
∂
γ 2 =2 γ ,γ =2 ∂ (∂⊥γ)ν+(∂⊤γ)τ , γ τ
∂t| u| h ut ui u t t | u|
=2∂⊥γ ∂ ν, γ (cid:10)τ +(cid:0)2γ ∂ ∂⊤γ (cid:1) (cid:11)
t h u | u| i | u| u t
= 2∂⊥γ γ k γ τ,τ +2γ ∂ ∂⊤γ
− t | u|h | u| i | u| u t
= 2k∂⊥γ γ 2+2(∂ ∂⊤γ)γ 2.
− t | u| s t | u|
Therefore
∂
ds=kk ds+(∂ ∂⊤γ)ds.
∂t ss s t
Using this we differentiate the integral to find
d d P
fds= f γ du
u
dt dt | |
Zγ Z0
P P
= f ds+ f(kk +∂ ∂⊤γ)ds+f(P,t)γ (P)P′
t ss s t | u |
Z0 Z0
= f +fkk f (∂⊤γ)ds.
t ss− s t
Zγ
We obtained the last equality using integration by parts and the periodicity of γ
with the identity
∂⊤γ(u,t) ∂⊤γ(u+P(t),t)= γ (u,t)P′(t),
t − t | u |
whichinturnfollowsfromthedefinitionofγ. (Noteinparticularthatthetangential
velocity ∂⊤γ is not periodic.) (cid:3)
t
ON THE CURVE DIFFUSION FLOW OF CLOSED PLANE CURVES 7
3. Curvature estimates in L2 and the isoperimetric ratio
The evolution equation (CD) is particularly natural as solutions decrease in
length while keeping enclosed area fixed. This is only necessarily true for curves
immersed in R2, and in fact this is the chief reason why we consider plane curves
as opposed to curves in Rn or immersed in a manifold.
Lemma 3.1. Suppose γ :S1 [0,T) R2 solves (CD). Then
× →
d d
L= k2ds, and A=0.
dt − s dt
Zγ
In particular, the isoperimetric ratio decreases in absolute value with velocity
d 2I
I = k2ds.
dt −L s
Zγ
Proof. Lemma 2.4 with f 1 gives
≡
d
L= kk ds= k2ds,
dt ss − s
Zγ Zγ
where we used integration by parts and the periodicity of the curve. For the area,
we first note that
(4) τ =kν, ν = kτ, ν =k τ (∂⊤γ)kτ.
s s − t sss − t
The first two relations are immediate from differentiating τ,ν =0 and using the
h i
definition of the curvature. For the third, we first compute the commutator of the
arc-length and time derivatives:
∂ =∂ γ −1∂ = γ −1∂ γ −2 ∂ γ ∂
ts t u u u tu u t u u
| | | | −| | | |
=∂s(cid:0)t+k ∂t⊥γ(cid:1)∂s− ∂s∂t⊤γ ∂s (cid:0) (cid:1)
(5) =∂st−k(cid:0)kss∂s(cid:1)− ∂s(cid:0)∂t⊤γ ∂s(cid:1).
(cid:0) (cid:1)
Using this and the first two equalities in (4) we compute the evolution of the unit
tangent vector field τ.
∂ τ =∂ γ
t ts
=∂ (∂⊥γ)ν+(∂⊤γ)τ kk γ ∂ ∂⊤γ γ
s t t − ss s− s t s
=−k(cid:0)sssν−kssνs+(∂(cid:1)t⊤γ)γss−kkss(cid:0)γs (cid:1)
(6) = k ν+(∂⊤γ)γ .
− sss t ss
Noting that ν 2 = 1 implies ν has no normal component, we obtain the final
t
| |
equality in (4) by differentiating ν,τ
h i
∂ ν,τ = ν,∂ τ = ν, k ν+(∂⊤γ)γ
h t i −h t i − − sss t ss
=ksss−k(∂t⊤γ).(cid:10) (cid:11)
8 GLENWHEELER
Returning to the area functional, we can now directly evaluate the derivative.
d 1 d
A= γ,ν ds
dt −2dt h i
Zγ
1
= k + γ,ν + γ,ν kk (∂⊤γ) γ,ν ds
−2 − ss h ti h i ss− t h si
Zγ
1
= γ,k τ k(∂⊤γ)τ + γ,ν kk (∂⊤γ) γ, kτ ds
−2 sss − t h i ss− t h − i
Zγ
1 (cid:10) (cid:11)
= k γ,τ k + γ,ν kk ds
ss s ss ss
−2 − −h i h i
Zγ
=0,
where we used Lemma 2.4 with f = γ,ν in the second line and integration by
h i
parts, the periodicity of γ and the formulae (4) throughout.
(cid:3)
We now turn our attention to the scale-invariantquantity
2
K =L k k ds,
osc
−
Zγ
(cid:0) (cid:1)
where
1
k = kds.
L
Zγ
Note that we have (and will continue to) suppressedthe dependence of K and L
osc
onγ(,t). Whenwemustindicate the dependence ofK andLonγ(,t), weshall
osc
· ·
use the notation K (t)=K γ(,t) and L(t)=L γ(,t) .
osc osc
· ·
A fundamental observation is that Lemmas 3.1 and 2.2 together imply K
osc
(cid:0) (cid:1) (cid:0) (cid:1) ∈
L1([0,T)).
Lemma 3.2. Suppose γ :S1 [0,T) R2 solves (CD). Then
× →
K <L4(0)/16π2.
osc 1
k k
Proof. Applying Lemma 2.2 with f =k k and recalling Lemma 3.1 we have
−
L3 1 d
K k 2 = L4,
osc ≤ 4π2k sk2 −16π2dt
so
t L4(0)
K dτ . (cid:3)
osc ≤ 16π2
Z0
The above lemma holds regardless of initial data, and appears to indicate that
the quantity K is a natural ‘energy’ for the flow.
osc
Remark3.3. Asimilarargumentasabovealsoshowsthat k 2 L1([0,T))with
k sk2 ∈
the estimate k 2 L(0). Although we will not need this fact, it does suggest
kk sk2k1 ≤
that k 2 is another well-behaved quantity under the flow.
k sk2
There exists an ω R satisfying
∈
(7) kds =2ωπ.
Zγ (cid:12)t=0
(cid:12)
Inthe casewherethe solutionisafamil(cid:12)yofclosedcurves,ω is thewinding number
(cid:12)
of γ(,0). Since the solution is a one-parameterfamily of smooth diffeomorphisms,
·
ON THE CURVE DIFFUSION FLOW OF CLOSED PLANE CURVES 9
andthewindingnumberisatopologicalinvariant,thewindingnumberofthecurves
γ(,t) remains constant. This can also be directly proven as in the lemma below.
·
Lemma 3.4. Suppose γ :S1 [0,T) R2 solves (CD) and
× →
kds =2ωπ.
Zγ (cid:12)t=0
(cid:12)
(cid:12)
Then (cid:12)
kds=2ωπ.
Zγ
In particular, the average curvature increases in absolute value with velocity
d 2ωπ
k = k 2.
dt L2 k sk2
Proof. Differentiating ν 2 =1 and τ 2 =1 gives that ν ,τ =0 and ν,τ =0.
t s ss
| | | | h i h i
Using this and (4), (6), we compute the evolution of the curvature as
∂
k =∂ ν,γ = ∂ ν,γ + ν,∂ γ
t ss t ss t ss
∂t h i h i h i
= ν,∂ τ
t s
h i
= ν,∂ τ kk τ ∂ ∂⊤γ τ
s t− ss s− s t s
=−(cid:10) k2kss−k ∂s∂t⊤γ (cid:0)+hν,∂(cid:1)sτti(cid:11)
=−k2kss−hν(cid:0),∂s(kss(cid:1)sν)i+(∂t⊤γ)hν,γsssi
(8) = k k2k +(∂⊤γ)k .
− ssss− ss t s
Therefore, applying Lemma 2.4 with f =k we have
d
kds= k +k2k k2k +(∂⊤γ)k (∂⊤γ)k ds=0,
dt − ssss ss− ss t s− t s
Zγ Zγ
using integration by parts and the periodicity of γ. This completes the proof. (cid:3)
We now compute the evolution of K .
osc
Lemma 3.5. Suppose γ :S1 [0,T) R2 solves (CD). Then
× →
d k 2
K +K k sk2 +2L k 2
dt osc osc L k ssk2
=3L (k k)2k2ds+6kL (k k)k2ds+2k2L k 2.
− s − s k sk2
Zγ Zγ
10 GLENWHEELER
Proof. This is a direct computation.
d
K = k2ds (k k)2ds+2L (k k) k k2k +(∂⊤γ)k ds
dt osc − s − − − ssss− ss t s
Zγ Zγ Zγ
(cid:0) (cid:1)
+L kk (k k)2 2k (k k)(∂⊤γ)ds
ss − − s − t
Zγ
k 2
= K k sk2 2L k 2+2L k2k2ds+4L k(k k)k2ds
− osc L − k ssk2 s − s
Zγ Zγ
L (k k)2k2ds 2L k(k k)k2ds
− − s − − s
Zγ Zγ
k 2
= K k sk2 2L k 2+4L k2k2ds 2kL kk2ds
− osc L − k ssk2 s − s
Zγ Zγ
L (k k)2k2ds.
− − s
Zγ
Rearranging,we have
d k 2
K +K k sk2 +2L k 2
dt osc osc L k ssk2
=4L k2k2ds 2kL (k k)k2ds 2k2L k2ds L (k k)2k2ds
s − − s − s − − s
Zγ Zγ Zγ Zγ
=3L (k k)2k2ds+6kL (k k)k2ds+2k2L k2ds.
− s − s s
Zγ Zγ Zγ
This proves the lemma. (cid:3)
AlthoughK isaprioricontrolledinL1,weneedmuchfinercontrolonK be-
osc osc
forewecanassertcontrolonothercurvaturequantitiesanddeduceglobalexistence.
(Indeed,globalexistenceisnottrueintheclassofsolutionsgivenbyTheorem2.1.)
While K is small, we do have the desired control. The following proposition
osc
gives us a pointwise estimate.
Proposition 3.6. Suppose γ : S1 [0,T) R2 solves (CD). If there exists a T∗
× →
such that for t [0,T∗) we have
∈
4π+24π2ω2 8π√3π√ω2+3πω4
K (t) − =2K∗,
osc
≤ 3
then during this time the estimate
t k 2
K +8ω2π2logL+ K k sk2dτ K (0)+8ω2π2logL(0)
osc osc osc
L ≤
Z0
holds.
Proof. Lemma 3.4, Corollary 2.3 and Ho¨lder’s inequality implies
3L
3L (k k)2k2ds K k 2,
− s ≤ 2π osck ssk2
Zγ
and
6kL (k k)k2ds 6ωL K k 2.
− s ≤ osck ssk2
Zγ
p