A Note on Grayson’s theorem

A note on Grayson's theorem

ANNIBALEMAGNI(*) - CARLOMANTEGAZZA(**)

ABSTRACT- In this note we show a variational proof of Matthew Grayson's con- vexification theorem for simple closed curves moving by curvature in the plane.

MATHEMATICSSUBJECTCLASSIFICATION(2010). 53C44, 35K93.

KEYWORDS. Curve shortening flow, geometric measure theory.

Contents

1 - Introduction. . . 263 2 - Analytical and Geometrical Facts . . . 265 3 - The Density Function. . . 267 4 - Blowing-Up at a Singularity - The Proof of Grayson's Theorem 269 References. . . 278

1. Introduction

Letg:S¹[0;T)!R²be the curvature flow of a simple smooth closed curve in the Euclidean plane, that is, satisfying@tgˆkn, such thatTis the maximal time of smooth existence. In this paper we give a new proof of the following result.

(*) Indirizzo dell'A.: Albert-Ludwigs UniversitaÈt Freiburg, Eckerstr. 1, Freiburg im Breisgau, Germany, 79104.

E-mail: [email protected]

(**) Indirizzo dell'A.: Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126, Pisa, Italy.

E-mail: [email protected]

THEOREM 1.1 (Grayson's Theorem [11]). At some time the evolving curve becomes convex.

REMARK 1.2. After the curve has become convex, the work of Gage and Hamilton [8, 9, 10] shows that it remains convex and it shrinks to a point ofR² in finite time, becoming asymptotically circular.

We borrow some ideas and techniques from the work of Ilmanen [16], Stone [20] and White [21]. The main line of the proof is quite standard, as we study the asymptotic behavior of the flow in presence of a singularity, which (as it follows from the comparison principle) must happen in finite time. This kind of analysis is usually performed by means of a parabolic blow-upof the flow by a dilation factor (at every time) which diverges with the same order as the maximum of the curvature. This allows one to extract from the rescaled flow a sequence of curves with bounded curvature, and to get ``easily'' a subsequence converging to some limit curve containing the information on the asymptotic shape. Accordingly to the blow-up rate of the maximum of the curvature ast approaches the singular timeT, the possible singularities are then subdivided into in the so-calledtype Iand type IIclasses. Precisely, if there exists a constantCsuch that

maxgt k² C

2(Tÿt) 8t2[0;T);

then the singularity is called of type I, if such constant does not exist, the singularity is called of type II.

Our analysis differs from the usual one insofar as we rescale the flow by the dilating factor 1=2(Tÿt) (which is the ``natural'' one for type I singu- larities), no matter what the rate of blow-up of the curvature is. Although this procedure gives no control on the maximum of the curvature along the blow-up, it allows to fully exploit Huisken's monotonicity formula [13]

(Theorem 2.3) to get anL²_loc-estimate on the curvature of a sequence of rescaled curves of the flow. This latter estimate implies the C_loc¹ -con- vergence of a subsequence either to a straight line through the origin ofR² or to an embedded, closed, convex curve with positive curvature (actually to the unit circle in R²). We will prove that, either by a theorem of White [21] (Theorem 4.6) or as a consequence of an application of the in- terior estimates of Ecker and Huisken [6] (Theorem 2.4), the convergence to the straight line is not possible. On the other hand, if the sequence of dilated curves converges to the unit circle inR², we will be able to exhibit a further sequence of curves in the rescaled flow which converges in the

C²-topology to an embedded, closed, convex curve with positive curvature, hence, such curves are definitely convex, thus proving Theorem 1.1.

We underline that the key fact, which allows us to pursue this ``unitary'' line of analysis without distinguishing the singularities into two types, is that in dimension one the uniform L²_loc-control on the curvature implies the C_loc¹ -subconvergence of the curves to some limit curve which, by a bootstrap argument starting from itsC¹-regularity, can then be shown to be smooth.

The lack of such functional embedding (and thus the lack of the pos- sibility to infer the smoothness of the limit along this line) is the main reason why our analysis is difficult to be pursued in higher dimensions, where the control of the mean curvature in L²_loc is not strong enough to give the C¹_loc-convergence of a subsequence of hypersurfaces to a limit.

Nevertheless, very interesting results in this direction have been obtained by Ilmanen in dimension two [16] (which is, in some sense, the critical case). In particular, assuming the mean convexity of the surfaces, Ilmanen shows that it is possible to get a smooth limit for a suitable sequence of rescaled surfaces of the flow.

REMARK1.3. Several techniques of this paper have been used by the authors, in collaboration with M. Novaga, also in the case of a network of three curves, concurrent at a triple point, moving by curvature to show that singularities cannot develop, see [18].

2. Analytical and Geometrical Facts

For any smooth, embedded, closed curveg:S¹!R²in the Euclidean plane, we will denote withs its arclength parameter and withnits inner unit normal vector field. The curvature of the curve will bekˆDd²g

ds²;nE , whereh;iis the Euclidean scalar product ofR².

DEFINITION2.1. Let T>0andg:S¹[0;T)!R² be a smooth one- parameter family of closed and embedded curves. We say thatgis a cur- vature flow of the initial curve g₀ˆg(;0) if for any time t2[0;T)and a2S¹there holds

@tg(a;t)ˆk(a;t)n(a;t):

For every t2[0;T), we will use the notationg_t:S¹!R²for the curve given byg_t(x)ˆg(x;t), that is, the evolving curve at time t2[0;T).

In this section we collect some well known results about the curvature flow gof an initial closed and embedded curve g₀ inR² (see [19], for in- stance, for more details and proofs).

PROPOSITION2.2. If the initial curveg₀is smooth, the curvature flow problem has a unique smooth solution defined on a maximal time interval[0;T)with T<‡ 1and, at every time t2[0;T), the curve g_t is a smooth embedding ofS¹inR².

The following evolution equations hold

@_tdsˆ ÿk²ds;

@tkˆkss‡k³ and

@tksˆksss‡4k²ks: Moreover, there holds

maxg_t k² 1

2(Tÿt) 8t2[0;T): (2:1)

Huisken's monotonicity formula [13] holds.

THEOREM2.3. For every x₀2R²we have d

dt Z

g_t

e^{ÿjxÿx4(Tÿt)0j2}



4p(Tÿt)

p dsˆ ÿ Z

g_t

e^{ÿjxÿx4(Tÿt)0j2}



4p(Tÿt)

p k‡hxÿx₀jni 2(Tÿt)

²ds (2:2)

for any t2[0;T).

The following interior estimates were proven by Ecker and Huisken in [6], see also [14].

THEOREM2.4. Let e1and e2be an orthonormal basis ofR²with respect to the standard Euclidean scalar product, let he1i ˆspanfe1g and he₂i ˆspanfe₂g. For x₀2R²and R>0, suppose that in a ball B_2R(x₀)the curve g_t, for t2[0;t), is a graph of a function over he₁i and let vˆ hnje₂i^ÿ1>0at time tˆ0.

SettingW(x;t)ˆR²ÿ jxÿx0j²ÿ2t, ifW_‡denotes the positive part of W, we have

v(x;t)W_‡(x;t)sup

x2g₀v(x;0)W_‡(x;0) (2:3)

for every t2[0;t)and x2g_t, as long as v(x;t)is defined everywhere on the support ofW_‡.

For arbitraryu2[0;1)we have the estimate

gt\BsupuR(x0)k²C(1ÿu)^ÿ2 1 R²‡1

t

BR(xsup0)[0;t)v⁴ (2:4)

for all t2[0;t), where the constant C is independent of t andg_t. Finally, we state a geometric result.

THEOREM 2.5 (Huisken's embeddedness measure [15]). For every t2[0;T), let L_t be the length of the curve g_t. We consider the function F_t :g_tg_t!Rgiven by

Ft(x;y)ˆ

pjxÿyj

L_t =sinpd_t(x;y)

L_t if x6ˆ y;

1 if xˆy;

8<

:

where d_t(x;y)is thegeodesicdistance between x and y alongg_t.

Then, we define the following infimum, which, in the case of closed curves, is actually a minimum:

E(t)ˆ inf

x;y2g_tF_t(x;y):

If the initial closed curveg₀ is embedded, the function E(t)is uniformly bounded below by a positive constant depending only on g₀, for every t2[0;T).

Since the function E(t)is positive if and only ifg_t is embedded, an initial embedded and closed curve remains embedded during all the flow.

3. The Density Function

As in Stone [20], we consider the followingGaussian density function.

For everyx₀2R², we define u(x0;t)ˆ

Z

gt

e^{ÿjxÿx4(Tÿt)0j2}



4p(Tÿt) p ds:

By Huisken's monotonicity formula (2.2), we have

@

@tu(x₀;t)ˆ ÿ Z

gt

e^{ÿjxÿx4(Tÿt)0j2}



4p(Tÿt)

p k‡hxÿx0jni 2(Tÿt)

²ds0 for every t2[0;T). Hence, the limit U(x₀)ˆlim

t!Tu(x₀;t) exists for every x₀2R²and, integrating, we get

u(x₀;0)ÿU(x₀)ˆ Z^T

0

Z

g_t

e^{ÿjxÿx4(Tÿt)0j2}



4p(Tÿt)

p k‡hxÿx0jni 2(Tÿt)

²ds dt: (3:1)

Letting m(t)ˆ sup

x02R²

u(x0;t), we can see that, as g_t is compact, this su- premum is actually a maximum for anyt2[0;T). Moreover, being a maxi- mum of a family of positive and monotone nonincreasing functions,m(t) is also monotone nonincreasing and positive. By looking at the right side of the monotonicity formula (2.2), we can finally notice that the functionsu(x₀;) are uniformly locally Lipschitz in time and we can conclude that the function mis locally Lipschitz as well, hence differentiable at almost everyt2[0;T).

Since m is monotone, we can define Sˆlim

t!Tm(t)2R^‡, which clearly satisfiesSU(x0) for everyx02R².

For everyt2[0;T) letx_t 2R²be a point at which m(t)ˆ

Z

g_t

e^{ÿjxÿ4(Tÿt)xtj2}



4p(Tÿt)

p ds

that is, x_t is a maximum point for the function u(;t):R²!R^‡. We now make use of the following lemma to compute the derivative of the function m.

LEMMA 3.1 (Hamilton's Trick [12]). Let M a Riemannian manifold and u:M(0;T)!Rbe a C¹ function such that for every time t, there exists a valued>0and a compact subset KMn@M such that at every time t⁰2(tÿd;t‡d)the maximum u_max(t⁰)ˆmax

p2Mu(p;t⁰) is attained at least at one point of K.

Then, umax is a locally Lipschitz function in (0;T)and at every dif- ferentiability time t2(0;T)we have

dumax(t)

dt ˆ@u(p;t)

@t

where p2M is any inner point such that u(;t)gets its maximum at p.

PROPOSITION3.2. For every t2[0;T), let xt2R² be as above, then

m⁰(t)ˆ ÿ Z

g_t

e^{ÿjxÿ4(Tÿt)xtj2}



4p(Tÿt)

p k‡hxÿx_tjni 2(Tÿt)

²ds (3:2)

at almost every t2[0;T). Moreover,

m(0)ÿSˆ Z^T

0

Z

g_t

e^{ÿjxÿ4(Tÿt)xtj2}



4p(Tÿt)

p k‡hxÿx_tjni 2(Tÿt)

²ds dt:

PROOF. Beingu(x;t) aC¹function, Hamilton's trick applies. Hence, at every differentiability timet2[0;T), by formula (2.2) we have

m⁰(t)ˆ ÿ Z

gt

e^{ÿjxÿ4(Tÿt)xtj2}



4p(Tÿt)

p k‡hxÿx_tjni 2(Tÿt)

²ds:

As the functionmis locally Lipschitz, we can integrate its derivative (which exists at almost every time) on every closed interval [0;Tÿe] and sendinge

to zero we get the second equality. p

4. Blowing-Up at a Singularity - The Proof of Grayson's Theorem As remarked in the introduction, we now rescale the curvature flow g:S¹[0;T)!R² in its maximal time interval [0;T) without distin- guishing between different types of singularities.

DEFINITION4.1. We define the rescaled flow as follows, eg(a;r)ˆg(a;t(r))ÿxt(r)

2(Tÿt(r))

p rˆr(t)ˆ ÿ1

2log (Tÿt);

obtaining a smooth flow of embedded, closed curveseg_r:S¹!R², defined for r2h

ÿ1

2logT;‡1 .

We rescale also formula (3.2) in order to get information on these curves. At almost everyr2h

ÿ1

2logT;‡1

it holds

d

drm(t(r))ˆm⁰(t(r))2(Tÿt(r))

ˆ ÿ Z

gt(r)

e^{ÿjxÿ4(Tÿt(r))xt(r)j2}



4p(Tÿt(r))

p k 

2(Tÿt(r))

p ‡hxÿxt(r)jni 2(Tÿt(r)) p

²ds

ˆ ÿ 1

p2p Z

~ g_r

e^ÿjyj22ek‡ hyjeni²ds0: Hence, integrating as before onh

ÿ1

2logT;‡1

, we get m(0)ÿSˆ 1

p2p Z

~ g_ÿ1

2 logT

e^ÿjyj22dsÿ lim

r!‡1

1

p2p Z

~g_r

e^ÿjyj22ds (4:1)

ˆ 1

p2p Z^‡1

ÿ¹₂logT

Z

~gr

e^ÿjyj22ek‡ hyjeni²ds dr<‡1:

REMARK 4.2. Thanks to inequality (4.1), for every family of disjoint intervals (ai;bi)h

ÿ1

2logT;‡ 1

such that P

i2N(biÿai)ˆ ‡ 1, we can find a sequencer_i2(a_i;b_i) such that

i!1lim 1

p2p Z

~g_ri

e^ÿjyj22ek‡ hyjeni²dsˆ0 (4:2)

and

i!1lim 1

p2p Z

~g_ri

e^ÿjyj22dsˆlim

i!1m(t(r_i))ˆS: (4:3)

Clearly, the sequenceriis monotone increasing and diverges to‡ 1.

We now state a technical lemma due to Stone which is crucial in order to commute limits and integrals.

LEMMA4.3 (Stone [20]). Let B_Ra ball of radius R>0inR², then the following estimates hold uniformly in r for the family of curves feg_rg_r2[ÿ¹_2logT;‡1).

1) There exist a constant C independent of BRsuch thatH¹(eg_r\BR) Ce^R2=2whereH¹is the one-dimensional Hausdorff measure inR².

2) For anye>0there exists a radius RˆR(e;Length(g₀);T)such that Z

~grnBR(0)

e^ÿjyj22dse:

PROPOSITION4.4. Let r_ibe a sequence as in Remark4.2. Then, either for every R>0the curvesg_ridefinitely stay outside of the ball B_R, or there exists a smooth, complete, embedded curveeg₁(either closed or unbounded) satisfyingek‡ heg₁jeni ˆ0, such that for a subsequence of ri(which we do not relabel) there holdsg_ri!eg₁in the C¹_loc-topology.

PROOF. Assume that we are in the second case. From the limit (4.2) and the estimate on the local length of the previous lemma, it follows that the sequence of curveseg_rihas curvatureslocallyequibounded inL², as the termjhyjenij²is clearly bounded in every ball ofR². Hence, we can extract a subsequence (not relabelled) that, after a possible reparametrization (in a constant multiple of the arclength of every curve), converges in C_loc¹ to a complete limit curveeg₁. Such curve satisfiesek‡ heg₁jeni ˆ0, as the in- tegral R

~g e^ÿjyj22ek‡ hyjeni²dsis lower semicontinuous under theC_loc¹ -con- vergence. The smoothness on the curveeg₁follows by a bootstrap argument applied to such relation.

Finally, the curve eg₁ is embedded as the Huisken's embeddedness measureEin Theorem 2.5 is scaling invariant and upper semicontinuous under the C_loc¹ -convergence of curves, hence, such quantity is bounded below by a positive constant also for the limit curveeg₁, which then must be

embedded. p

By the classification result of Abresch-Langer and Epstein-Weinstein (see [1] and [7] or [17, 19]), the limit curveeg₁can be either a line through the origin or the unit circle centered in the origin ofR². Actually, for our purposes, we only need the weaker conclusion that the limit curve is either a line through the origin or a closed, convex curve with positive curvature.

LEMMA4.5. Any smooth, complete, embedded curvesinR²satisfying k‡ hsjni ˆ0is either a line through the origin ofR²or a closed, convex curve with positive curvature.

PROOF. The relation kˆ ÿhsjniimplies that the curvature satisfies the ODEk_sˆkhsjti, wheretˆ@_ssis a unit tangent vector. Suppose that

at some point it holdskˆ0. By the ODE, at the same point holds alsoks ˆ0 and hence, by the uniqueness theorem for ODE's, we conclude thatkis identically zero and we are dealing with a lineL which, ashxjni ˆ0 for everyx2L, must contain the origin ofR².

We can now suppose thatkis always different from zero and, possibly after a reversion of the orientation of the curve, we can assume thatk>0 at every point and thus that the curve is strictly convex.

The derivative ofjsj²with respect to the arclength parameter it is easily computed to be

@sjsj²ˆ2hsjti ˆ2ks=kˆ2@slogk:

Thus we getkˆCe^jsj2=2, for some constantC>0 and it follows thatkis bounded from below byC>0.

We now consider a new coordinate aˆarccoshe₁jni which, by the convexity of the curve, is a good global parametrization (obviously, as for the arclength parameters, the functionais only locally continuous since it

``jumps'' after a complete round).

Differentiating with respect to the arclength parameter we have

@_saˆkand

kaˆks=kˆ hsjti kaaˆ@ska

k ˆ1‡khsjni

k ˆ1

kÿk:

Multiplying both sides of the last equation by 2ka we get

@a[k²_a‡k²ÿlogk²]ˆ0, which means that the quantityk²_a‡k²ÿlogk²is equal to some constantZalong all the curve. Notice that such quantityZ cannot be less than 1 and that, ifZˆ1, thenkmust be constant and equal to one along the curve, which consequently must be the unit circle cen- tered at the origin ofR².

If Z>1, it follows that k is uniformly bounded from above. Hence, recalling thatkˆCe^jsj2=2, the image of the curve is contained in a ball ofR² and, by the embeddedness and completeness hypotheses, the curve must be closed, simple and strictly convex, ask>0 at every point. p

Since the second point of Lemma 4.3 implies that

i!1lim 1

p2p Z

~g_ri

e^ÿjyj22dsˆ 1

p2p Z

~g₁

e^ÿjyj22ds;

and, by equation (4.3), the left hand side is equal toS, we deduce that if we have a non empty limit curve thenS1 and, by direct computation, we

conclude that ifS>1 theneg₁cannot be a line through the origin and hence it must be a closed, simple curve with positive curvature.

We now see thatS must be larger than one. To this aim we start with defining the set ofreachable pointsof the curvature flowgby

S ˆ fx2R²j 9 ti% Tandai2S¹ such thatg_ti(ai)!xg: It is easy to see thatSis non empty and compact. Moreover, ifx₀62 S, the flow is clearly definitely far fromx₀. If insteadx₀2 S, for everyt2[0;T) the closed ball of radius 

2(Tÿt)

p and centerx₀intersectsg_t. Indeed, let dtˆmin_a2S¹jW(a;t)ÿx0j, that is, the Euclidean distance from x0 to the curveg_t.

The functiondt :[0;T)!Ris obviously locally Lipschitz and at a dif- ferentiability time withd_t>0, by Hamilton's trick, we have

d⁰_t(x)ˆ@

@tjg_t(b)ÿx0j k(b;t)hn(b;t)jg_t(b)ÿx0i jg_t(b)ÿx₀j for anyb2S¹such thatd_tˆ jg_t(b)ÿx₀j.

By minimality, the closed ballBdt(x0) intersects the curveg_tonly on its boundary and the vector g_t(b)ÿx0

jg_t(b;t)ÿx₀jis parallel to the normaln(b;t). An easy geometric argument shows then that

k(q;t)hn(q;t)jg_t(b)ÿx0i

jg_t(b)ÿx0j ÿ1=dt:

Thus we conclude that, for almost every timet2[0;T) and if d_t6ˆ0, we have

d⁰_t(x) ÿ1=d_t:

Integrating this differential inequality on the time interval [t;t] we get d²_t ÿd²_t 2(tÿt) and, by the hypothesis onx₀, we haved²_ti!0, hence

d²_t ˆlim

i!1d²_t ÿd²_ti lim

i!12(tiÿt)ˆ2(Tÿt); which proves our claim.

Chosenx₀2 S, we rescale the flow as follows, this time around the fixed pointx₀,

g(a;r)ˆg(a;t(z))ÿx₀ 2(Tÿt(z))

p zˆz(t)ˆ ÿ1

2log (Tÿt);

obtaining a smooth flow of embedded, closed curvesg_z:S¹!R², defined for z2h

ÿ1

2logT;‡1

. Notice that, by the above discussion, all these curves intersect the unit closed ball ofR².

Rescaling also formula (3.1), we have u(x₀;0)ÿU(x₀)ˆ 1

p2p Z^‡1

ÿ¹₂logT

Z

gz

e^ÿjyj22k‡ hyjni²ds dz<‡1: Then, repeating the arguments in Remark 4.2 and Proposition 4.4, there exists a subsequence ofz_i(which we do not relabel) such that the curvesg_zi converge in theC_loc¹ -topology to a smooth, complete, embedded curveg₁, which is not empty (the limit curve also must intersect the unit closed ball of R²), satisfiesk‡ hg₁jni ˆ0 and

1

p2p Z

g1

e^ÿjyj22dsˆlim

i!1

1

p2p Z

g_zi

e^ÿjyj22dsˆU(x₀):

Hence, by Lemma 4.5 and a direct computation, it follows thatU(x₀)1, henceS1. Moreover, if we assume thatSˆ1 we conclude thatU(x0)ˆ1 for everyx02 Sand the subsequence of rescaled curvesg_ziconverges in the C¹_loc-topology to a line through the origin ofR².

The following general local regularity theorem of White [21] (holding for the mean curvature flow in any dimension) can now be applied.

THEOREM4.6. There exists a constante>0such that if a point x02 S satisfies U(x0)<1‡e, then there exists a radius R>0 such that in B_R(x₀)[0;T)R²Rthe curvature is uniformly bounded.

Clearly, this theorem contradicts the assumptionSˆ1, as (by a com- pactness argument on the setS) it implies that the curvature is uniformly bounded whent!Tand this is impossible since estimate (2.1) holds.

In the special case of simple curves, the exclusion of the caseSˆ1 can be actually deduced as follows also by means of the interior estimates of Ecker and Huisken (Theorem 2.4).

Chosen anyx02 S, by theC_loc¹ -convergence of the rescaled curveseg_zito a line through the origin ofR², for everyR>2 there is a sequence of times t_i%Tand a line L passing forx0such that every curveg_tiis a graph over L in the ballB_R 

2(Tÿti)

p (x₀).

Supposing that x0ˆ0 and that L is he1iin R², the pieces of curves g_t\B_R 

2(Tÿti)

p can be represented, at least for small times after ti, as a graphs of a time dependent function. Moreover, choosingilarge enough, for every x2g_ti\B_R 

2(Tÿti)

p , the quantityv(x;t)ˆ hn(x;t)je₂i^ÿ1 is arbi- trarily close to one at timetˆt_i.

As the circle @B 

2(T‡eÿt)

p is moving by curvature and, choosinge>0 small enough, at time tˆt_i it is contained in the ball B_R 

2(Tÿti)

p , by the Sturmian theorem of Angenent in [4, Proposition 1.2] and [3, Section 2]

(see [2] for the proof), we have that the number of intersections (which is two at timeti) between the curveg_t and the circle@B 

2(T‡eÿt)

p cannot in-

crease in time. On the other hand, the number of intersections can also not decrease, otherwise the whole curveg_twould be definitely contained inside B 

2(T‡eÿt)

p , in contradiction with the fact that the limit of the rescaled curves eg_zi is the unbounded line L. This argument shows that it is not possible that other parts of the moving curve g_t ``get into'' the ball B 

2(T‡eÿt)

p at some time t>t_i. Consequently, the only reason for which g_t\B 

2(T‡eÿt)

p can possibly stop being a graph is that the tangent vector to such graph becomes vertical at some time, or, equivalently, that the functionvis not bounded.

Inequality (2.3) excludes this fact if the quantityv at time ti is small enough. Hence, with a suitable choice of one of the times ti, by esti- mate (2.4), the curvature of g_t for t2[t_i;T) is bounded in the ball B 

2(T‡eÿt)

p , in particular it is bounded in B 

p2eB 

2(T‡eÿt)

p for every

t2[t_i;T).

By the same argument above, this implies that the curvature is uni- formly bounded ast!T, contradicting inequality (2.1).

REMARK 4.7. We underline that the key point in getting a bound on the curvature by means of this argument is theC_loc¹ -convergence of the rescaled curves to a line (by theL²bound on the curvature), which does not follow in higher dimensions.

Once established thatS>1, getting back to the original rescaling as in Proposition 4.4, we can assume that the sequence of rescaled curveseg_ri converges inC_loc¹ to a closed, convex curveeg₁inR²with positive curvature.

Hence, as the curveeg₁is compact, the convergence is actually inC¹ with equibounded curvatures inL².

Grayson's Theorem is then a consequence of the following proposition, which clearly implies that the curves g_t(qi) definitively have positive cur- vature and hence are convex.

PROPOSITION 4.8. There exists a sequence q_i% ‡1 such that the rescaled curveseg_qi converge in the C²-topology to a closed, convex curve with positive curvature.

PROOF. Fixingi2Nand lettingrˆrÿr_i<1, asrˆ ÿ1

2log (Tÿt), we compute

d dr

Z

~g_r

(ek²‡rek²_s)dsˆ2(Tÿt)d dt

Z

g_t



2(Tÿt)

p k²ds‡

Z

~g_r

ek²_sds

‡2(Tÿt)rd dt

Z

g_t

( 

2(Tÿt)

p )³k²_sds

ˆ ÿ 

2(Tÿt)

p Z

g_t

k²ds‡( 

2(Tÿt)

p )³

Z

g_t

(2kkss‡k⁴)ds

‡ Z

~ g_r

ek²_sdsÿ3( 

2(Tÿt)

p )³r

Z

g_t

k²_sds

‡( 

2(Tÿt)

p )⁵r

Z

g_t

(2ksksss‡7k²k²_s)ds

ˆ Z

~g_r

hÿek²‡2ekekss‡ek⁴‡ek²_sÿ3rek²_s‡2rekseksss‡7rek²ek²_si ds;

where we used the formulas in Proposition 2.2.

Integrating by parts and by Peter-Paul inequality, we have Z

~ g_r

ek²ek²_sdsˆ1 3 Z

~g_r

@_s(ek³)ek_sdsˆ ÿ1 3 Z

~ g_r

ek³ek_ssds1 6 Z

~ g_r

ek⁶‡ek²_ssds

and d dr

Z

~ gr

(ek²‡rek²_s)ds Z

~gr

hÿek²_s‡ek⁴ÿek²ÿ3rek²_sÿ2rek²_ss‡7r(ek⁶‡ek²_ss)=6i ds

Z

~gr

(ÿek²_s‡ek⁴‡3rek⁶)ds:

Thus, the following interpolation inequalities, holding for any closed curve of lengthLin the plane (see Aubin [5, page 93]),

kekk⁴_L4Ckek_sk_L2kekk³_L2‡C

Lkekk⁴_L2 and kekk⁶_L6Ckek_sk²_L2kekk⁴_L2‡C L²kekk⁶_L2; imply (also by means of the Young inequality)

Z

~gr

ek⁴ds1 2 Z

~gr

ek²_sds‡1 2

Z

~gr

ek²ds

!₃

‡ Z

~gr

ek²ds

!₃

‡ C L³(eg_r) and

3r Z

~g_r

ek⁶ds r Z

~ g_r

ek²_sds

!₃

‡2 Z

~g_r

ek²ds

!₃

‡ C L²(eg_r)

Z

~g_r

ek²ds

!₃ :

Hence, as we know thatL(eg_r)R

~ gr

e^ÿjyj22ds 

p2p

andr<1, we conclude

d dr

Z

~gr

(ek²‡rek²_s)ds Z

~gr

(ÿek²_s‡ek²_s=2)ds‡C Z

~gr

ek²ds

!₃

‡C

‡ r Z

~gr

ek²_sds

!₃

‡C Z

~gr

ek²ds

!₃

C Z

~gr

ek²ds

!₃

‡ r

Z

~gr

ek²_sds

!₃

‡C

C Z

~ gr

(ek²‡rek²_s)ds

!₃

‡C;

for a constantCindependent ofrr_iandi2N.

Integrating this differential inequality for the quantity Q_i(r)ˆ R

~ gr

(ek²‡(rÿri)ek²_s)dsover the interval [ri;ri‡2d], it is easy to see that if d>0 is small enough, we have

Q_i(r)C(d;Q_i(r_i))ˆC d;

Z

~g_ri

ek²ds

!

ˆC(d);

for everyr2[ri;ri‡2d], as the curveseg_rihave uniformly bounded curva- ture inL². Hence, ifr2[ri‡d;ri‡2d] we have the estimate

Z

~gr

(ek²‡dek²_s)ds Z

~ gr

(ek²‡(rÿr_i)ek²_s)dsC(d)

which implies

Z

~ g_r

(ek²‡ek²_s)dsC(d)‡C(d) d ;

for everyr2[ri‡d;ri‡2d] and a constantC(d) independent ofi2N.

We can now find as before, using again Remark 4.2, a sequence of valuesq_i2[r_i‡d;r_i‡2d] such that

i!1lim 1

p2p Z

~g_qi

e^ÿjyj22ek‡ hyjeni²dsˆ0:

and

i!1lim 1

p2pZ

~g_qi

e^ÿjyj22dsˆ lim

i!1m(t(q_i))ˆS>1:

As this new sequence of rescaled curveseg_qi satisfies the local length es- timate of Lemma 4.3 andekandek_sare uniformly bounded inL², arguing as in Proposition 4.4 and in the subsequent discussion, we can extract a subsequence (not relabelled) that converges inC² to a limit curve which cannot be a line and hence, by Lemma 4.5, must be a closed curve with

positive curvature. p

Acknowledgments. Annibale Magni was partially supported by the DFG Collaborative Research Center SFB/Transregio 71. Carlo Mante- gazza was partially supported by the Italian FIRB Ideas ``Analysis and Beyond''.

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Manoscritto pervenuto in redazione il 9 Aprile 2013.