Singularities of the area preserving curve shortening flow with a Neumann free boundary condition

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by  Elena Mäder-Baumdicker

SINGULARITIES OF THE AREA PRESERVING CURVE
SHORTENING FLOW WITH A FREE BOUNDARY CONDITION
ELENAMA¨DER-BAUMDICKER
Abstract. WeconsidertheareapreservingcurveshorteningflowwithNeumann
7
freeboundaryconditionsoutsideofaconvexdomainoratastraightline.Wegive
1
acriteriononconvex initialcurves thatguarantees theappearance ofatypeII
0
2 singularityinfinitetime. Anappropriaterescalingatthefinitemaximaltimeof
existenceyieldsagrimreaperorhalfagrimreaperaslimitflow. Weconstruct
n
examplesofinitialcurvessatisfyingthementionedcriterion.
a
J

1
3
1. Introduction

]
G Theareapreservingcurveshorteningflow(APCSF)forclosedplanecurveswas
D introduced byM.Gagein1986[6]. Itisthe“steepest descendflow”forthelength
. functional under the constraint that the enclosed area is constant. For a family of
h
simpleclosedcurvesγ : S1×[0,T) → R2,theevolution equationturnsouttobe
t
a
m d κds 2π
γ = κ− ν = κ− ν,
[ dt  R L  L !
1 where we use the following notation: ν = Jτ is the normal of the curves, where
v J is the rotation by +π; κ is the curvature with respect to ν, L is the length of
8 2
4 the curves and ds denotes integration by arclength. M. Gage proved in [6] that a
9 strictly convexsimpleclosedcurveremainsstrictly convexundertheAPCSF.The
8
curves converge for t → ∞smoothly to acircle enclosing the same enclosed area
0
asγ . Thus,theflowconverges tothesolution oftheisoperimetric problem inR2.
. 0
1
This problem consists in finding the shortest closed curve enclosing a fixed area.
0
7 Theanalogresultforn-surfacesinRn+1,n ≥ 2wasprovedbyG.Huiskenin[9]: A
1 uniformly convex, embedded surface moving according to the volume preserving
v: meancurvatureflowstaysuniformlyconvexandexistsforalltimest ∈ [0,∞). The
i moving surfaces converge smoothly toasphere enclosing the samevolume as the
X
initialsurface.
r
a Weconsider theAPCSFinafreeboundary setting andwanttoknowwhenand
how singularities develop. Butatfirst werecall what isknown about singularities
intheclosedsituation.
J. Escher and K. Ito considered in [5] immersed closed curves possibly with
self-intersections. Then the evolution equation is dγ = κ− 2πm ν where m ∈ Z
dt L
is the index (or turning number) of the immersed closed(cid:16)curves.(cid:17)The index m is
independent of time. Escher and Ito proved that an immersed curve with m ≥ 1
andenclosedareaA < 0orm ≥ 2andL2 < 4πmA developsasingularityinfinite
0 0 0
time. TheproofisinspiredbytheworkofK.-S.Chouonthesurfacediffusionflow
KarlsruheInstitute ofTechnology,Institute forAnalysis, Englerstr. 2, 76131Kalrsruhe,
Germany
E-mail address: [email protected].
1
2 E.MA¨DER-BAUMDICKER
for curves [2]. X.-L. Wang and L.-H. Kong also studied immersed closed curves
movingaccordingtotheAPCSF[16]. Theyprovedthattheflowexistsforalltimes
andconvergessmoothlytoanm-foldcirclewhentheinitialcurveisconvexandhas
so-called “n-fold rotational symmetry” and index m (n > 2m). Onthe other hand,
“Abresch-Langertype”curveseitherconvergetoamultiplecoverofacircle(when
A > 0) or the curvature blows up at finite time (when A < 0) or the curvature
0 0
blows upat themaximal time ofexistence (when A = 0), see [16, Theorem 1.2].
0
Notethatthereareexamples whereonlyaslightchange isnecessary todeform an
initialcurvewithA < 0intoonewith A = 0andthenintoonewithA > 0.
0 0 0
Wenow explain the free boundary setting of the APCSFwhich was studied by
the author in [12, 11]. Let Σ ⊂ R2 be a convex simple closed curve in the plane
and orient it positively. Wecall Σ the support curve. It is not moving in time. An
initialcurveγ : [a,b] → R2 isacurvethatsatisfiesγ (a),γ (b) ∈ Σand
0 0 0
π π
(1) ∠ τ (a),Σ~τ(γ (a)) = , ∠ τ (b),Σ~τ(γ (b)) = − ,
R2 0 0 R2 0 0
2 2
(cid:16) (cid:17) (cid:16) (cid:17)
where ∠ denotes the oriented angle in R2, Σ~τ is the tangent of Σ and τ is the
0
tangent ofγ . Theseconditions meanthatthecurveγ startsinΣinto theexterior
0 0
domain with respect to Σ and comes back from the exterior. The contact angles
are 90 degree angles. Wenow let the curve γ flow according to the APCSFsuch
0
that these conditions are preserved, i.e. γ : [a,b] ×[0,T) → R2 satisfies (1) and
γ(a,t),γ(b,t) ∈ Σforallt ∈ [0,T)and
d κds
γ = κ− ν.
dt  R L 
Asthecurvesarenotclosedthequantity κdsisnotanintegertimes2πingeneral.
It is in fact the first step to find conditioRns that guarantee a bound of κ¯ := κds
R L
independent of t. In [12], the author proved that the flow in this setting does not
developasingularity whentheinitialcurvesatisfiesfourconditions:
i) γ isstrictlyconvex,
0
ii) itisembedded,
iii) itiscontained intheexteriordomainwithrespecttoΣand
iv) itsatisfies L < 4 arcsin A0 ,
0 5max|Σκ| L2
(cid:18) 0(cid:19)
where A istheenclosed areaofthedomainenclosed byγ andthepartofΣcon-
0 0
necting γ (b) and γ (b). Furthermore, the curves γ(·,t) subconverge smoothly for
0 0
t → ∞toanarcofacirclesittingoutside ofΣandmeetingΣperpendicularly.
Inthispaperweanswerthefollowingquestionsthatnaturallyarisewhenstudy-
ingthissetting:
• Can convex curves develop a singularity under the APCSF in the free
boundary setting?
• Dotheyappearinfinitetime?
• Ofwhattypearethesingularities?
• Howdoesablowupatthesingulartimelooklike?
Forourmaintheoremwehavetointroduce somenotation. Letγ :[a,b] → R2 be
0
an initial curve with κ > 0. Define σ˜ as the connection of γ (a) to γ (b) along
0 0 0 0
Σ (with the positive orientation of Σ). In the case γ (a) = γ (b) we choose σ˜ to
0 0 0
SINGULARITIESFORACONSTRAINTCURVATUREFLOW 3
be one full circulation of Σ. The assembled curve γ −σ˜ is a closed curve with
0 0
exterior angles +π. Then we call the index of γ − σ˜ , l := ind(γ − σ˜ ) ∈ Z,
2 0 0 0 0
the loop number of γ . Define Σd ≔ min{|x− y| : x,y ∈ Σ,Σ~τ(x) = −Σ~τ(y)} and
0
let A(γ ,σ˜ ) be the oriented enclosed area of γ −σ˜ . The enclosed area of Σ is
0 0 0 0
denotedby A(Σ). Withthisnotation wecanstateourmaintheorem:
Theorem1.1. Letγ : [a,b] →R2beaninitialcurvewithloopnumberlsatisfying
0
Σ
κ > 0and L < d. Weconsider threedifferentsituations:
0 0
i) Either∠ Σ~τ(γ (a)),Σ~τ(γ (b)) ∈ (0,π)andγ satisfies A(γ ,σ˜ )> 0and
R2 0 0 0 0 0
L20 ≤ π(cid:16)(2l−1)2. (cid:17)
A(γ0,σ˜0) l
ii) Or∠ Σ~τ(γ (a)),Σ~τ(γ (b)) ∈ (π,2π]andγ satisfiesA(γ ,σ˜ )+A(Σ)> 0
R2 0 0 0 0 0
and (cid:16) L20 ≤ π(2l+1)2.(cid:17)
A(γ0,σ˜0)+A(Σ) l+1
iii) Or∠ Σ~τ(γ (a)),Σ~τ(γ (b)) ∈ (0,π)and A(γ ,σ˜ )< 0.
R2 0 0 0 0
Inanyofthesec(cid:16)asesthesolutionof(cid:17)theareapreservingcurveshorteningflowwith
Neumann free boundary conditions outside of Σ develops a singularity in finite
time,i.e.T < ∞. Thesingularity isoftypeIIinthesensethat
max
sup |κ|(p,t) → ∞(t → T )and
max
p∈[a,b]
sup |κ|2(p,t)(T −t) isunbounded.
max
p∈[a,b](cid:16) (cid:17)
The“Hamiltonblow-up” atT yieldseither agrim reaperwithout boundary or
max
halfagrimreaperatastraightline.
Remark i) The “Hamilton blow-up” was defined in [7]. We will explain it
intheproofofCorollary2.19.
ii) There is numerical evidence given by U. F. Mayer [13] that there are em-
bedded closed curves that at first get a self-intersection and then develop
a singularity under the APCSF. In the free boundary setting, it seems to
bethecasethatthereareinitiallyembeddedcurvesthatstayembeddedbut
developasingularityinfinitetime. Theseexamplesofinitialcurvessatisfy
conditionii)fromTheorem1.1,seeExampleThreeinSection3. Wethink
thatthesecurvesdevelopasingularity attheboundary.
Wealsostudythesituation atastraightline. Theresultisasfollows.
Theorem 1.2. Let γ : [a,b] → R2 be an initial curve at a straight line Σ with
0
κ > 0. Let δ be the closed curve getting by reflecting γ at Σ. Let ind(δ ) =: m
0 0 0 0
be the index of δ . Then m = 2k+1, k ∈ N∪{0}. And the area preserving curve
0
shortening flow with Neumann free boundary conditions at the line Σ develops a
typeIIsingularity infinitetimeifoneofthefollowingconditions issatisfied:
i) Either A(δ ) < 0.
0
ii) Orm ≥ 3and L(δ )2 < 4πmA(δ ).
0 0
The structure of this paper is as follows. In Section 2 we recall definitions and
some results from [11]that are necessary for the proof of Theorem 1.1. Then, we
prove uniform bounds 0 < c ≤ κ¯(t) ≤ c < ∞ and L(γ(·,t)) ≥ c > 0 under
1 2 3
the conditions of Theorem 1.1. These bounds imply with [12] subconvergence to
a part of a circle that is possibly (partly) multicovered. Together with the results
from[12]wegetthatthatthesingularity isoftypeII.
4 E.MA¨DER-BAUMDICKER
InSection3,wegiveexamplesofcurvesthatdosatisfytheconditions ofTheo-
rem1.1andthusdevelopatypeIIsingularity infinitetime.
Section4contains theproof ofTheorem 1.2. WereflectthecurvesatthelineΣ
andapplytheresultsfrom[5]. Wecombinethiswithresultsfrom[12]toshowthat
thesingularity isoftypeII.
Acknowledgment
TheauthorwouldliketothankJonasHirschforveryusefuldiscussions.
2. SingularitiesoftypeIIinfinitetime
Notations Let f : [a,b] → Rn, n ∈ {1,2}, be a C1-map, f = f(p). Let γ :
[a,b] → R2 be a piecewise smooth, regular curve. We denote by ∂ f ≔ 1 ∂ f
s |∂pγ| p
the derivative with respect to arclength of f. We define ds ≔ |∂ γ|dp. We recall
p
theformulaforthecurvature ofγ
κ(p) = h∂2γ(p),ν(p)i,
s
where ν = Jτ = J∂ γ is the normal of the curve γ, J is the rotation by +π in the
s 2
plane.
Definition 2.1. We call a smooth, regular, convex, simple and smoothly closed
curve Σf : [Σa,Σb] → R2 a support curve. We assume Σf to be parametrized by
arclength. We choose the orientation of the curve in the way that Σκ ≥ 0. We use
thenotation
Σ ≔ Σf [Σa,Σb] .
(cid:16) (cid:17)
ConsidertheperiodicextensionofΣf andcallitΣf˜:R → R2. Weusethenotation
a2ΣκdsΣ fortheintegralofthecurvature ΣκofΣf˜froma ∈ Rtoa ∈ R.
a1 1 2
BRy the Jordan curve theorem, the curve Σ separates R2 into a bounded and an
unbounded domain. Thebounded domainisenclosed byΣandisdenoted byGΣ.
We define Σd ≔ min{|x − y| : x,y ∈ Σ,Σ~τ(x) = −Σ~τ(y)}, the smallest distance
betweentwoparallellinesinR2 thattouchGΣ.
Definition 2.2. A planar, smooth, regular curve γ : [a,b] → R2 is called initial
0
curveifitsatisfiestheconditions
γ (a),γ (b) ∈ Σ
0 0
π
∠ τ (a),Σ~τ(γ (a)) = ,
R2 0 0
2
(cid:16) (cid:17)
π
∠ τ (b),Σ~τ(γ (b)) = − ,
R2 0 0
2
(cid:16) (cid:17)
whereτ = ∂ γ isthetangent ofγ andΣ~τisdefinedby
0 s 0 0
Σ~τ: Σ → S1 ⊂ R2, Στ= Σ~τ◦Σf,
where Στ is the tangent of Σ. The symbol ∠ (v,w) denotes the oriented angle
R2
betweentwovectorsv,w ∈ R2.
SINGULARITIESFORACONSTRAINTCURVATUREFLOW 5
Definition2.3. Letγ : [a,b] →R2beaninitialcurve. Asmoothfamilyofsmooth,
0
regularcurvesγ :[a,b]×[0,T) → R2 thatsatisfies
∂γ
(p,t) = (κ(p,t)−κ¯(t))ν(p,t) ∀(p,t) ∈ [a,b]×[0,T),
∂t
γ(p,0) = γ (p) ∀p∈ [a,b],
0
(2) γ(a,t),γ(b,t) ∈ Σ ∀t ∈ [0,T),
π
∠ τ(a,t),Σ~τ(γ(a,t)) = ∀t ∈ [0,T),
R2
2
(cid:16) (cid:17)
π
∠ τ(b,t),Σ~τ(γ(b,t)) = − ∀t ∈ [0,T)
R2
2
(cid:16) (cid:17)
iscalledasolutionoftheareapreservingcurveshorteningproblemwithNeumann
freeboundary conditions. Here,κ¯ denotestheaverage ofthecurvature,
κ(p,t)ds κ(p,t)ds
κ¯(t)≔ = .
R ds RL(γ(·,t))
R
Remark For a smooth initial curve, the existence and uniqueness of the solution
of (2) is standard. One gets short time existence on a short time interval [0,T ].
0
The solution can be extended up to a maximal time of existence T ≤ ∞. By
max
regularity theoryforparabolic Neumannproblemsthecurvessatisfy
γ ∈C2+α,1+α2 [a,b]×[0,Tmax),R2 ∩C∞ [a,b]×(0,Tmax),R2 , α ∈(0,1),
(cid:16) (cid:17) (cid:16) (cid:17)
where C2+α,1+α2 denotes the usual parabolic Ho¨lder space. If Tmax < ∞ then
max |κ|(·,t) → ∞ (t → T ). A source for the existence for closed curves
[a,b] max
moving by a geometric flow with a constraint is for example [4]. The technique
how to transform the free boundary problem into a standard Neumann boundary
problem can be found in [14, 15]. For our specific situation a sketch of the exis-
tenceandregularity resultisin[12,Proposition 2.4].
Definition 2.4. Let Σ be a support curve and let γ : [a,b] → R2 be a curve with
γ(a),γ(b) ∈ Σ. Then we call a curve σ : [a˜,b˜] → Σ ⊂ R2 with σ(a˜) = γ(a) and
σ(b˜) = γ(b)aboundary curveonΣwithrespecttoγ.
Definition 2.5. Let γ : [a,b] × [0,T) → R2 be a solution of (2). Consider a
C1-family of smooth curves σ : [a˜,b˜] × [0,T) → Σ with σ(a˜,t) = γ(a,t) and
σ(b˜,t) = γ(b,t) for all t ∈ [0,T), i.e. σ ≔ σ(·,t) is a boundary curve on Σ with
t
respect to γ ≔ γ(·,t). Then for each t ∈ [0,T), we call the following expression
t
theorientedareaenclosed byγ andΣ:
t
1 1
(3) A(γ,σ)≔ p1dp2− p2dp1 − p1dp2 − p2dp1.
t t
2Z 2Z
γt σt
Werecallsomebasicproperties provedin[12].
Lemma2.6(Lemma2.6,Lemma2.11andCorollary 2.14[12]). Letγ : [a,b] →
0
R2 be a smooth initial curve. Then we have the following properties: The area
preserving curve shortening flow is curve shortening and area preserving, i.e.
dL(γ) ≤ 0 and dA(γ,σ) = 0 on [0,T), where γ : [a,b] × [0,T) → R2 is a
dt t dt t t
solution of (2)and σ : [a˜,b˜]×[0,T) → Σ is aC1-family of boundary curve on Σ
withrespecttoγ.
6 E.MA¨DER-BAUMDICKER
AsthedomainGΣ isconvexandweare“outside”ofΣtheflowimprovesconvexity
tostrictconvexity. Thisis,κ ≥ 0fortheinitialcurveimpliesκ > 0on[a,b]×(0,T).
0
Remark Ifγ isasmooth initialcurve thenaC1-familyofboundary curvesσon
0
Σ with respect to γ exists. This was proved in [12, Lemma 2.9]. In the case that
κ > 0wewillalsoshowthisinLemma2.13.
0
Wenowstartwithprovingsomebasicgeometricproperties ofourcurves.
Definition 2.7. Let f : [a,b] → R2 be a piecewise smooth, regular and closed
curve. Thenumber
ind(f) ≔n(∂ f,0) ∈ Z
p
is called the index (or turning number) of f. Here, n(∂ f,0) denotes the winding
p
numberofthecurve∂ f :[a,b] → R2 withrespectto0 ∈ R2.
p
Proposition 2.8. LetΣ ⊂ R2 beapositively oriented convex smoothJordan curve
andletγ : [a,b] → R2 beaC2-curvewithκ > 0and
γ(a),γ(b) ∈ Σ,
π
∠ τ(a),Σ~τ(γ(a)) = ,
R2
2
(cid:16) (cid:17)
π
∠ τ(b),Σ~τ(γ(b)) = − .
R2
2
(cid:16) (cid:17)
Thenwehavethat κds ≥ π. Furthermore, ifγ(a) = γ(b)then κds ≥ 3π.
R R
Proof. In [12, Proposition 3.1], it was shown that the geometric situation of the
curves imply κds ≥ π. Thestatement therewasformulated forasolution of(2).
Buttheonly pRroperties ofthecurves thatareused intheproof arestrict convexity
andtheboundary conditions. Ifγ(a) = γ(b)then
(4) κds > π
Z
was proved in [12, Proposition 3.1] as well. We improve this to κds ≥ 3π. By
assumption, γ is a closed piecewise smooth curve with exterior aRngle π at γ(a) =
γ(b). Bytheindextheorem (seeTheoremA.1)wegetthat
1 1
Z ∋ k := ind(γ) = κds+ .
2π Z 2
We use (4) to get π < κds = 2kπ−π which implies k > 1. Thus, we have that
k ≥ 2and κds = 2kπR−π ≥ 3π.
(cid:3)
R
Lemma 2.9. Let γ : [a,b] → R2 be an initial curve with κ > 0. Define the
0 0
“simplest”boundarycurveσ˜ onΣwithrespecttoγ astheconnectionofγ (a)to
0 0 0
γ (b) along Σ (with the positive orientation of Σf). In the case that γ (a) = γ (b)
0 0 0
then we choose σ˜ to be one full circulation of Σ. With these definitions we have
0
that
(5) ind(γ −σ˜ ) =:l∈ N,
0 0
whereγ −σ˜ denotes thecurve whichisassembled byγ and (inreversed direc-
0 0 0
tion)σ˜ .
0
SINGULARITIESFORACONSTRAINTCURVATUREFLOW 7
Proof. LetΣf˜: R → ΣbetheperiodicextensionofthesupportcurveΣf anddefine
a ≔Σf−1(γ (a)),
0 0
Σf−1(γ (b)) ifΣf−1(γ (b)) > a
b ≔ 0 0 0
0 ( Σf−1(γ0(b))+(Σb−Σa) ifΣf−1(γ0(b)) ≤ a0.
This definition makes sure that a < b ≤ a + (Σb − Σa). Now the “simplest”
0 0 0
boundary curveonΣwithrespecttoγ is
0
σ˜ (p) ≔ Σf˜((b −a )p+a ) for p∈ [0,1].
0 0 0 0
Thiscurveistheconnectionofγ (a)toγ (b)alongΣ,inparticularwiththepositive
0 0
orientation ofΣf˜. Itissmooth andregular. Notethat inthecase γ (a) = γ (b) the
0 0
boundarycurveσ˜ isexactlyoneperiodofΣf˜,thusσ˜ isclosedand κ ds =
0 0 σ˜0 σ˜0 σ˜0
b0ΣκdsΣ = 2πinthis case. Ingeneral, thecurve σ˜ satisfies 0 ≤ bR0ΣκdsΣ ≤ 2π
a0 0 a0
bRecauseofa < b ≤ a +(Σb−Σa). Nowwedefineγ˜ = γ −σ˜ :R[a,b+1] → R2
0 0 0 0 0 0
by
γ (p) for p ∈ [a,b],
γ˜ (p) ≔ 0
0 Σf˜(−(b −a )(p−b)+b ) for p ∈ (b,b+1]×[0,T).
 0 0 0
This is the closedcurve assembled by γ and in reversed direction σ˜ . The curve
0 0
γ˜ ispiecewisesmooth, regularandhasexteriorangles π.
0 2
Bytheformulafortheindex(seeTheoremA.1inAppendixA)wehave
ind(γ˜0) = 1 κds|t=0 − 1 b0ΣκdsΣ + 1 ≕ l∈ Z.
2π Z 2π Z 2
a0
Weprovel∈ N. Ifl ≤ −1then
b0Σ Σ
κds|t=0 ≤ κds −3π ≤ 2π−3π < 0,
Z Z
a0
acontradiction toκ > 0,hencel≥ 0. Butweevenhavel ≥ 1: Ifl = 0then
0
(6) π ≤ κds|t=0 = −π+ b0ΣκdsΣ ≤ −π+2π = π,
Z Z
a0
where we used κds|t=0 ≥ π from Proposition 2.8. The boundary conditions
imply that thereRcannot be any curvature of Σ between γ (a) and γ (b): Since
0 0
γ0(a) = γ0(b) would imply κds|t=0 ≥ 3π (due to Proposition 2.8) we have that
γ (a) , γ (b). AfterrotationRand translating weare inthe situation γ (a) = 0and
0 0 0
γ0(b)−γ0(a) = −e . Because of the convexity and the positive orientation of Σ, we
|γ0(b)−γ0(a)| 1
have
Σ
hγ(b)−γ(a), ν(γ(a))i ≥ 0.
We hence get hΣ~τ(γ(a)),e i ≥ 0 and hΣ~τ(γ(b)),e i ≤ 0. The boundary conditions
2 2
imply hτ(a),e1i ≥ 0 and hτ(b),e1i ≥ 0. The only way to get κds|t=0 = π is that
τ (a) = e and τ (b) = −e , every other angle of the tangentRwould imply more
0 2 0 2
contribution to the angle κds|t=0. But by the convexity of Σ the line segment
between γ (a) and γ (b) mRustbe inΣ, soweare inthe situation ofFigure 1. This
0 0
8 E.MA¨DER-BAUMDICKER
Σ
γ (b) γ (a)
0 0
Figure1. Thecase κds = π
R
impliesthattheboundarycurveσ˜ isthestraightlinebetweenγ (a)andγ (b). By
0 0 0
theformulafortheindex,wethenhave
1 1
ind(γ˜0)= κds|t=0 + = 1
2π Z 2
whichcontradicts l = 0.
(cid:3)
Definition 2.10. Let γ : [a,b] → R2 be an initial curve with κ > 0 on [a,b].
0 0
Then wecall l ≔ ind(γ −σ˜ ) the loop number of γ , where σ˜ is the “simplest”
0 0 0 0
boundary curveonΣwithrespecttoγ dueLemma2.9. Wehaveprovedl∈ N.
0
Lemma2.11. Letγ : [a,b] → R2 bean initial curve withκ > 0on [a,b]. Then
0 0
thereisaboundary curveσ :[0,1] → ΣonΣwithrespecttoγ suchthat
0 0
ind(γ −σ ) = +1,
0 0
whereγ −σ denotes thecurve whichisassembled byγ and (inreversed direc-
0 0 0
tion)σ .
0
Proof. In Lemma 2.9 we showed that the loop number l of γ satisfies l ∈ N. In
0
general, l can be big. We now define a boundary curve on Σ with respect to γ
0
that compensates high contribution of κds|t=0 to the index l. If l = ind(γ˜0) = 1
we are done and define σ ≔ σ˜ . So wRe assume l ≥ 2. Now we define σ to be
0 0 0
theboundarycurvewhichstartsinγ (a)andendsinγ (b)butcirculatestheclosed
0 0
curveΣ(l−1)-manytimes:
σ (p) ≔ Σf˜ (b +(l−1)(Σb−Σa)−a )p+a , p ∈ [0,1].
0 0 0 0
(cid:16) (cid:17)
Thenγˆ = γ −σ isregularbecause a < b ≤ b +(l−1)(Σb−Σa). Theclosed,
0 0 0 0 0 0
piecewise smooth curve γˆ has exterior angles +π. We use the formula for the
0 2
index(TheoremA.1),toget
ind(γˆ0)= 1 κds|t=0 − 1 b0+(l−1)(Σb−Σa)ΣκdsΣ + 1
2π Z 2π Z 2
a0
= 21π Z κds|t=0 − 21π Z b0ΣκdsΣdsγˆ0 + 12 − 21π(l−1)2π
a0
= ind(γ˜ )−(l−1) = 1.
0
(cid:3)
SINGULARITIESFORACONSTRAINTCURVATUREFLOW 9
Lemma2.12. Letγ : [a,b] → R2beaninitialcurvewithκ > 0andloopnumber
0 0
l ∈ N. Letγ : [a,b]×[0,T) → R2 bethesolution of(2). Recall
a ≔ Σf−1(γ (a)),
0 0
Σf−1(γ (b)) ifΣf−1(γ (b)) > a
b ≔ 0 0 0
0 ( Σf−1(γ0(b))+(Σb−Σa) ifΣf−1(γ0(b)) ≤ a0.
Thenthereexistuniquemapsa :[0,T) → Randb :[0,T) → Rsuchthat
a(0) = a andΣf˜(a(t)) = γ(a,t)∀t ∈ [0,T),
0
b(0) = b andΣf˜(b(t)) = γ(b,t)∀t ∈ [0,T).
0
ThesemapsareC1 on[0,T)andsmoothon(0,T).
Proof. Themapsa,b : [0,T) → Raretheliftsofγ(a,t)andγ(b,t)witha(0) = a
0
and b(0) = b , see [3, Chapter 5.6, Proposition 2]. Tobe very precise we get lifts
0
on[0,T−ǫ]foreveryǫ > 0. Byuniquenesstheliftsa,barewell-definedon[0,T).
The differentiability follows because γ(a,·) and γ(b,·) are C1 in zero and smooth
on(0,T). (cid:3)
Lemma2.13. Letγ : [a,b] → R2beaninitialcurvewithκ > 0andloopnumber
0 0
l ∈ N. Letγ : [a,b]×[0,T) → R2 bethesolution of(2). Define
σ(p,t)≔ Σf˜ (b(t)+(l−1)(Σb−Σa)−a(t))p+a(t) , p∈ [0,1],
wherea,b : [0,T) →(cid:16)Raretheliftsofγ(a,t)andγ(a,t)from(cid:17) Lemma2.12. Define
theassembledcurves
γ¯ ≔γ−σ.
Thenγ¯ hasthefollowing properties:
i) a(t) < b(t) +(l−1)(Σb−Σa) ∀t ∈ [0,T). This implies in particular that
γ¯(·,t)=:γ¯ isregular foreveryt ∈ [0,T).
t
ii) For all t ∈ [0,T), the curves γ¯ are piecewise smooth, regular, closed and
t
theyhaveexteriorangles π. Withrespecttot,γ¯ isC1 on[0,T)andsmooth
2
on(0,T).
iii) ind(γ¯ )= 1 ∀t ∈ [0,T).
t
iv) κds = (2l−1)π+ b(t)ΣκdsΣ ∀t ∈ [0,T).
a(t)
R R
Proof. Asl ∈ Nandthedefinitionofa ,b wegeta < b ≤ b +(l−1)(Σb−Σa).
0 0 0 0 0
Weassumethat thereisatimet ∈ (0,T)witha(t ) = b(t )+(l−1)(Σb−Σa)and
0 0 0
a(t) < b(t)+(l−1)(Σb−Σa)on[0,t ). Thenσ areregularfort ∈ [0,t ). Theindex
0 t 0
formula(Theorem A.1)implies
1 1 b(t)+(l−1)(Σb−Σa) 1
ind(γ −σ) = κds− ΣκdsΣ + ,
t t
2π Z 2π Z 2
a(t)
where we used that −σ is negatively oriented. We have ind(γ −σ ) = 1 due to
t 0 0
thedefinitionofγ andLemma2.11. Continuityoftheindexintimplies
t
b(t)+(l−1)(Σb−Σa)
(7) κds = π+ ΣκdsΣ fort ∈ [0,t ).
0
Z Z
a(t)
This yields κdst=t0 = π and we get a contradiction to Proposition 2.8 because
a(t ) = b(t )R+(l−1)(Σb−Σa)impliesγ(a,t ) = γ(b,t ). Thisshows(i). Nowthe
0 0 0 0
otherproperties followimmediately. (cid:3)
10 E.MA¨DER-BAUMDICKER
Remark We mention here that the statements from the beginning of Section 2
up to here were proved in [11, Section 3 & 4]. In Theorem 4.2.4 [11], a bound
0 < c ≤ κ¯(t) ≤ c < ∞wasshownundertheconditionT < ∞. Wewanttofind
1 2 max
conditions on the initial curve that guarantee a bound on κ¯ even if T = ∞. We
max
dothisinthefollowing.
Proposition 2.14. Let γ : [a,b] → R2 be an initial curve with κ > 0 and loop
0 0
number l ∈ N and L < Σd = min{|x − y| : x,y ∈ Σ,Σ~τ(x) = −Σ~τ(y)}. Let
0
γ : [a,b]×[0,T)bethesolutionof(2). Thentherearetwopossibilities.
i) Either∠ Σ~τ(γ (a)),Σ~τ(γ (b)) ∈ (0,π). Thenwehavethat
R2 0 0
(cid:16) (cid:17)
κds ∈ ((2l−2)π,2lπ)∀t ∈ [0,T).
Z
ii) Or∠ Σ~τ(γ (a)),Σ~τ(γ (b)) ∈ (π,2π]. Then
R2 0 0
(cid:16) (cid:17)
κds ∈ (2lπ,(2l+2)π)∀t ∈ [0,T).
Z
Proof. As|γ (a)−γ (b)| ≤ L < ΣdwegetΣ~τ(γ (a)) , −Σ~τ(γ (b)). ByLemma2.13
0 0 0 0 0
wehaveforeveryt ∈ [0,T)
b(t)
(8) κds = (2l−1)π+ ΣκdsΣ
Z Z
a(t)
Casei)meansthat∠ Σ~τ(γ (a)),Σ~τ(γ (b)) = b0ΣκdsΣ ∈ (0,π).Nowweget
R2 0 0 a0
(cid:16) (cid:17) R
b(t)
Σ Σ
(9) κds ∈ (−π,π)forallt ∈ [0,T).
Z
a(t)
We prove this by contradiction. If there is a t ∈ [0,T) with b(t)ΣκdsΣ < −π
a(t)
b(t)Σ Σ R
(or κds > π) then continuity implies that there must be a t < t such that
a(t) 0
b(t0R)ΣκdsΣ = −π(orπ). Butthisyields
a(t0)
R
∠ Σ~τ(γ(a,t ),Σ~τ(γ(b,t )) = −πor
R2 0 0
∠ (cid:16)Σ~τ(γ(a,t ),Σ~τ(γ(b,t ))(cid:17) = π.
R2 0 0
(cid:16) (cid:17)
InanycaseΣ~τ(γ(a,t )) = −Σ~τ(γ(b,t ))whichcontradicts
0 0
Σ
|γ(a,t )−γ(b,t )| ≤ L(γ ) ≤ L < d.
0 0 t0 0
In Case ii) wehave that ∠ Σ~τ(γ (a)),Σ~τ(γ (b)) = b0ΣκdsΣ ∈ (π,2π]. With (8)
R2 0 0 a0
wehavetoprove b(t)ΣκdsΣ(cid:16)∈ (π,3π)forallt ∈ ((cid:17)0,TR). ThisfollowsasinCasei).
a(t)
(cid:3)
R
Proposition 2.15. Letγ : [a,b]×[0,T ) → R2 be the solution of (2)where the
max
initial curve γ : [a,b] → R2 has loopnumber land satisfies κ > 0and L < Σd.
0 0 0
Furthermore, weassumethatγ satisfies
0
i) A(γ ,σ ) , −(l−1)A(Σ) if∠ Σ~τ(γ (a)),Σ~τ(γ (b)) ∈ (0,π)or
0 0 R2 0 0
ii) A(γ ,σ ) , −lA(Σ) if∠ Σ~τ(γ(cid:16)(a)),Σ~τ(γ (b)) ∈ (π(cid:17),2π].
0 0 R2 0 0
Thenthereisaconstant δ > 0such(cid:16)that L(γ) ≥ δforal(cid:17)lt ∈ [0,T ).
t max