RIGIDITY IN THE HARMONIC MAP HEAT FLOW. We establish various uniformity properties of the harmonic map heat æow,

Transcription

1 j. differential geometry 45 è1997è RIGIDITY IN THE HARMONIC MAP HEAT FLOW PETER MILES TOPPING Abstract We establish various uniformity properties of the harmonic map heat æow, including uniform convergence in L 2 exponentially as t!1, and uniqueness of the positions of bubbles at inænite time. Our hypotheses are that the æow is between 2-spheres, and that the limit map and any bubbles share the same orientation. 1. Introduction Let us consider smooth maps ç : S 2! S 2. We use z = x + iy as a complex coordinate on the domain, obtained by stereographic projection, and write the metric as ç 2 dzdçz, where çèzè= 2 1+jzj 2 : Similarly we have a coordinate u on the target, and a metric ç 2 dudçu. We are using the notation dz = dx + idy; dçz = dx, idy; with analogues for du and dçu, and we will write u z = 1 2 èu x, iu y è; u çz = 1 2 èu x + iu y è: To the map ç we associate the energy densities èçè = ç2 èuè ç 2 ju zj 2 ; èçè = ç2 èuè ç 2 ju çzj 2 ; Received October 30,

2 594 peter miles topping and eèçè èçè: The corresponding energies are èçè = èçè = and S 2 èçè = i 2 S 2 èçè = i 2 C C ç 2 ju z j 2 dz ^ dçz; ç 2 ju çz j 2 dz ^ dçz; è1è èçè: We also deæne a local energy E èx;rè èçè = eèçè; B r èxè where B r èxè is the geodesic ball of radius r centred at x in S 2. The Jacobian of ç is given by and consequently we see that Jèçè èçè, èçè; è2è èçè, èçè =4çdegèçè: For a æxed target chart, we may form ç ç = 4 çu ç 2 zçz + 2ç u è3è ç u zu çz The associated geometric object is and the tension of ç is to be T @çu ; which is a section of ç æ èts 2 è. The critical points of the energy functional E are known as harmonic maps, and the Euler-Lagrange equation which they satisfy is T èçè =0:

3 rigidity in the harmonic map heat flow 595 The harmonic map heat æow is a solution æ : S 2 æ ë0; 1è! S 2 of the associated parabolic = T èæè; æèæ; 0è = ç 0; which we refer to as the `heat equation'. We call ç 0 the `initial map'. The heat æowwas introduced by Eells and Sampson ë2ë. It is L 2 -gradient æow on the energy - loosely speaking, æ evolves in order to decrease its energy as quickly as possible. For æ çç S 2 we measure the concentration overæofaæow with the quantity EèR; æè = sup E èx;rè èæèæ;tèè: èx;tè2ææë0;1è We will have cause to embed the target S 2 in R 3, and see æ as a map v : R 2 æ ë0; 1è! S 2,! R 3,or^v : S 2 æ ë0; 1è! S 2,! R 3, and ç 0 as a map ^v 0 : S 2! S 2,! R 3. In terms of v we have and so Eèçè = 1 2 eèçè = 1 2 jrvj2 ; R 2 jrvj 2 dxdy: In the harmonic map heat æow, the map v evolves according = 1 ç 2 èæv + vjrvj2 è; where æ is the Laplace-Beltrami operator. We will denote the righthand side of è5è also by T, without confusion. We may also make references such aseèvèæ;tèè meaning Eèæèæ;tèè for related v and æ. Existence theory for the heat equation was studied by Struwe in ë6ë. The equation was shown to have a global weak solution èwhich is now known to be essentially unique ë3ëè and to be smooth except at ænitely many points in space-time. For much of this paper we will be considering only the asymptotic behaviour of the æow at inænite time, in which case we may assume it is globally smooth. Struwe's theory also described how `bubbles' may occur in the heat æow. We need an extension of this which is the following theorem of Qing ë4ë.

4 596 peter miles topping Theorem 1. Let v be the solution of è5è, corresponding to a solution æ of è4è. Then there exist ænitely many non-constant harmonic maps f^! k g m k=0 from S2 to S 2 èseen as maps f! k g m k=0 from R2 to S 2 via stereographic projectionè together with sequences èiè ft i g with t i!1, èiiè ffa k i ggm k=1 in R2 with lim i!1 a k = i xk 2 R 2 for 1 ç k ç m èwhere x k corresponds to a point ^x k 2 S 2 è, and èiiiè ffç k i ggm k=1 with çk i é 0 for 1 ç k ç m and any i, and lim i!1 ç k i = 0 for 1 ç k ç m; such that ç k i ç j i + çj i ç k i + jak i, aj i j2 ç k i çj i!1; as i!1; and è6è X m lim Eèvèæ;tèè = Eè! k è; t!1 k=0 and moreover, vèx; t i è, mx k=1 ç ç ç x, a k i ç! k,! ç k k è1è!! 0 i strongly in W 1;2 loc èr2 ; R 3 è as i!1. Remark 1. In fact, Qing proves more than Theorem 1. We are using his theorem on Palais-Smale-type sequences rather than his theorem on the harmonic map heat æow. Of course, to be able to do this we must ænd a sequence ft i g with t i!1such that T èæèæ;t i èè! 0in L 2 ès 2 è. Such a sequence is easy to ænd èsee ë6ë or deduce it from è16è.è Qing's work has now been generalised by Ding and Tian ë1ë. We will refer to the map ^! 0 as the `body map', and to the maps f^! k g m k=1 as `bubbles'. The points f^xk g m k=1 will be known as bubble points, or blow-up points. Of course, for è6è to hold, we should choose our domain chart so that none of the blow-up points correspond to the point at inænity in the domain, though this is just a technical point.

5 rigidity in the harmonic map heat flow 597 As the statement of Qing's theorem is technical, we will describe some of its implications in more intuitive language. The theorem says ærstly that there exists a sequence of times ft i g at which the heat æow ^v converges weakly in W 1;2 ès 2 ; R 3 è to the harmonic map ^! 0 èand in particular strongly in L p ès 2 ; R 3 è for p 2 ë1; 1èè and that we have the strong convergence è7è ^vèæ;t i è! ^! 0 in W 1;2 loc ès2 nfx 1 :::x m g; R 3 è as i!1: The theorem also says that near the bubble points fx k g m k=1, the energy of the æow concentrates, and that by rescaling appropriate regions by appropriate amounts, we see new maps - the bubbles. So much was known from the work of Struwe ë6ë. An important aspect of the theorem is that it tells us that all the energy of the æow is accounted for by the body map and the bubbles. In this paper we show that some of the asymptotic properties of the heat æow hold uniformly as t!1rather than just at a special sequence of times ft i g. 2. Statement of the results Before we state our results, we must recall that any harmonic map from S 2 to S 2 èor more generally from S 2 to a surfaceè is either holomorphic or anti-holomorphic. The proof is simple and may be found in ë8ë for example. We now give our main theorem. Theorem 2. Suppose we have a solution æ of the heat equation è4è, and the corresponding v and ^v. Suppose moreover that at inænite time, the bubbles and the body map are all holomorphic or all antiholomorphic. Then with the deænition of ^! 0 and f^x k g m k=1 as in Theorem 1 we have that: èiè ^vèæ;tè * ^! 0 uniformly as t!1weakly in W 1;2 ès 2 ; R 3 è - and hence strongly in L p ès 2 ; R 3 è for any p 2 ë1; 1è, èiiè ^vèæ;tè! ^! 0 uniformly as t!1in C k loc ès2 nf^x 1 :::^x m g; R 3 è for any k 2 N, èiiiè for any ré0 suæciently small and k 2f1 :::mg, the quantity E è^xk ;rèèæèæ;tèè converges to a limit F k;r uniformly as t!1.

6 598 peter miles topping Remark 2. In fact, we can control the rate of convergence too èsee ë7ëè. It will follow from the proofs of our theorems, for example, that given æ çç S 2 nf^x 1 :::^x m g there exist Cé0 and æé0 such that: èiè k^vèæ;tè, ^! 0 k L 2èS 2 è ç C è^vèæ;tèèè 1 2 ç Ce,æt, èiiè k^vèæ;tè, ^! 0 k W 1;2èæè ç C è^vèæ;tèèè 1 2 ç Ce,æt, èiiiè je è^xk ;rèèæèæ;tèè, F k;r jçc è^vèæ;tèèè 1 2 ç Ce,æt. A consequence of parts èiiè and èiiiè of Theorem 2 is that we cannot pick another sequence of times ft i g in Qing's theorem and get a diæerent set of blow-up points f^x k g. Although our result may be true if we allow some of the maps f! k g m k=0 to be holomorphic and others anti-holomorphic èin other words absolutely no restrictions on the æowè it is no longer true if we drop the condition that the target is S 2. An example in which parts èiè and èiiè fail is given later. In fact, we believe that part èiiiè may also fail -asketched counter-example will be given in ë7ë in which a diæerent sequence of times ft i g gives a diæerent number of bubbles. We remark that we dohave an example of a heat æow satisfying the hypotheses of Theorem 2 in which bubbling occurs. We will give this in ë7ë. The body map in this example is constant. We also have the following perturbation result for æows which are smooth for all time èi.e., æows with no bubbles at ænite time - see Struwe ë6ë.è Theorem 3. Suppose we have two solutions of the heat equation, which we write as maps ^v and ^w from S 2 æ ë0; 1è to S 2,! R 3, with initial maps ^v 0 and ^w 0 from S 2 to S 2,! R 3. Suppose moreover that for the æow ^v there are no bubbles at ænite time and that the bubbles and the body map at inænite time are all holomorphic or all anti-holomorphic. Then with f^x k g the blow-up points of the æow ^v as in Theorem 1, we have that for all "é0, æ çç S 2 nf^x 1 :::^x m g and ré0 suæciently small, there exists æ é 0 independent of ^w such that if then k^v 0, ^w 0 k W 1;2 ès 2 è éæ; èiè k^vèæ;tè, ^wèæ;tèk L 2èS 2 è é"for all té0, èiiè k^vèæ;tè, ^wèæ;tèk W 1;2 èæè é"for all té0,

7 rigidity in the harmonic map heat flow 599 In other words, if we start a new æow close to the original one, then it will stay close in L 2 for all time, and close in W 1;2 away from the blow-up points. We remark that the perturbed æow may blow up in ænite time. As in Theorem 2 the hypothesis that the target is S 2, rather than something higher dimensional, cannot simply be dropped. In fact in ë7ë we will give an example of a harmonic map èfrom S 2 to a higher dimensional targetè which blows up under arbitrarily small C 1 perturbations. By considering a perturbation of a harmonic mapping from T 2 to an equator of S 2,we see that the theorem is not true for general domain surfaces. Remark 3. The condition that the body map and bubbles at inænite time are all holomorphic or all anti-holomorphic will certainly be satisæed if we impose the condition Eèç 0 è ç 8ç +4çjdegèç 0 èj on the initial map ç 0. This follows from è1è and è2è and the fact that if ç : S 2! S 2 is a non-trivial holomorphic map, then èçè ç 4ç. We omit any further details. 3. The key estimate In this section we derive akey estimate controlling in terms of the tension. The estimate is very similar to the key estimate of Leon Simon in his important paper ë5ë. However, in the special case of maps between 2-spheres, and with the harmonic map energy functional E,we are able to reduce Simon's hypothesis of W 2;2 closeness to a harmonic map, to just smallness of This makes it applicable to maps with bubbles, assuming the bubbles and the body map are all anti-holomorphic èor all holomorphicè. Very loosely speaking, the harmonic map heat æow can only keep moving energy about for all time if the total energy is dissipated very slowly. The point of the estimate will be to show that this cannot happen, and thus that the heat æow becomes `rigid' for large times. Lemma 1. There exist " 0 é 0 and çé0 such that providing ç : S 2! S 2 satisæes èçè é" 0, we have the estimate è8è èçè ç çkt èçèk 2 L 2 ès 2 è :

8 600 peter miles topping Before proving Lemma 1 we recall the following lemma from ë9, Theorem 2.8.4ë. Lemma 2. Suppose we have an operator I on functions f : C! R given by èifèèwè= Then for q 2 è1; 2è we have the estimate kifk L C fèzè i dz ^ dçz: jz, wj 2 ç Cèqèkfk 2q L ècè: 2,q ècè q Proof. èlemma 1è. Fix global complex coordinates z and u on the domain and target respectively. With these coordinates, we consider the quantity ç 2 u z.to begin with, we calculate è9è è10è èç 2 u z è çz = ç 2 u zçz +2çç u u çz u z +2çç çu çu çz u z = 1 4 ç2 ç 2 ç +2çç çu ju z j 2 ; by è3è. In particular, as ç çu =, 1 2 uç2,we see that è11è jèç 2 u z è çz jçjç 2 ççj + ç 2 ju z j 2 : We now apply Cauchy's theorem for C 1 functions to the function ç 2 u z to get, for jwj ér, ç 2 u z èwè = 1 ç 2 u z z, w dz + 1 2çi Dr èç 2 u z è çz dz ^ dçz; z, w where D r = fz 2 C : jzj érg. Allowing r to tend to inænity, and observing that jç 2 u z j!0asjzj!1èbecause j ç ç u zj is boundedè we see that ç 2 u z èwè = 1 èç 2 u z è çz dz ^ dçz: 2çi z, w Combining this with è11è, we have jç 2 u z èwèj ç 1 ç C C 1 jz, wj èjç2 ççj + ç 2 ju z j 2 è i dz ^ dçz: 2 Now we observe that the right-hand side is independent of which stereographic chart we took for the target ènote for example that jççj is the

9 rigidity in the harmonic map heat flow 601 length of èsoby taking a chart for which 02 C corresponds to we obtain the estimate è12è jçu z èwèj ç 1 2ç C 1 jz, wj èjç2 ççj + ç 2 ju z j 2 è i dz ^ dçz: 2 But now, all the terms are independent of the target chart, allowing us to change target chart, or equivalently to move w with a æxed chart. To develop estimate è12è we need to appeal to the theory of Riesz potentials, and in particular to Lemma 2. This produces, for q 2 è1; 2è, the ærst of the estimates è13è è14è è15è kçu z k L ç, æ ç C 2q kç 2 ççk L q ècè + kç 2 ju z j 2 k L q ècè 2,q ècè ç C kççk L q ès2 è + kçu z k 2 L 2q ècè ç ç C kççk L2 ès 2 è + kçu z k L2 ècèkçu z k ç L ç 2q ; 2,q ècè whilst the last follows from applying Híolder's inequality to both terms. The constant C is changing, of course, but remains independent of ç. Now, as èçè =kçu z k 2 L 2 ècè,we see that there exists " 0 é 0 such that providing èçè ç " 0,wemay absorb the second term on the right-hand side into the left-hand side to give This then easily yields è S 2 + èçè! 1 2 kçu z k L ç CkT k 2q 2,q ècè L 2 ès 2 è: = kçu z k L2 èd 1 è ç Ckçu z k L ç CkT k 2q L 2,q èd1 è 2 ès 2 è; where S 2 + is the hemisphere corresponding to the points in the domain with jzj é 1. Repeating the estimate with the `opposite' chart to give estimate over the remaining hemisphere, and combining the two, we are left with èçè ç CkT k 2 L 2 ès 2 è : q.e.d.

10 602 peter miles topping Remark 4. We remark that we have in fact proved more than stated in that we can control the L p norm of èçè for any p ç 1 not just p =1. Remark 5. Our key lemma is the part of this work which requires the hypotheses on the domain, target and æow. As we shall see, for a æow æèæ;tè èæèæ;tèè is exactly half the energy still left to be dissipated during the æow, but in fact whenever we have an estimate of the form èenergy left to dissipateè çktk p L 2 ès 2 è for some pé1, the forthcoming proofs will be valid. Of course, such an estimate will not be true in general as the conclusions of our theorems are not true in general. Although we only discuss the case of round 2-spheres in this work, we mention that the key lemma as stated implies the same lemma with a deformed domain metric, and that the proof can be modiæed to imply the same lemma with a deformed target metric. 4. Proof of the results Before giving the proofs we state a supporting lemma which is a consequence of successive iterations of a result of Struwe ë6, Lemma ë. The lemma gives control of C k norms of the æow away from any points where the energy density concentrates. Lemma 3. There exists " 1 é 0 such that whenever we have a solution æ:s 2 æë0; 1è! S 2 of the heat equation è4è satisfying EèR; æè é" 1 for some Ré0 and æ çç S 2, then the Híolder norms of æ are bounded uniformly on æ æ ëç;1è for any çé0. Proof. ètheorem 2è. Without loss of generality, we will assume that the body map and all the bubbles are anti-holomorphic, rather than holomorphic. We begin by recalling the well known fact that è16è d dt Eèæèæ;tèè =,kt èæèæ;tèèk2 L 2 ès 2 è : This is easily proved by writing E in terms of v, and using è5è. Moreover, as a combination of è1è and è2è gives èçè = 1 2 èeèçè+4çdegèçèè;

11 we see that è17è rigidity in the harmonic map heat flow 603 d dt =, 1 2 kt èæèæ;tèèk2 L 2 ès 2 è : Qing's description of the bubble tree in Theorem 1, together with the fact that the maps f! k g are all anti-holomorphic, tells us that at his sequence of times ft i g, is converging to zero. As energy is decreasing èequation è17èè we then see that è18è èæèæ;tèè! 0 as t!1: In particular, there exists a time T such that if t ç T, then èæèæ;tèè é " 0, where " 0 is deæned in Lemma 1. As we are concerned only with the asymptotics of the heat æow, we may suppose for simplicity that T = 0. Moreover, we may assume that no ænite time blow-up occurs. We note then that the combination of Lemma 1 and equation è17è implies the exponential decay èæèæ;tèè which is necessary to establish the exponential convergence mentioned in Remark 2. An alternative application of Lemma 1 gives, d dt 1 2 = kt èæèæ;tèèk 2 L 2 ès 2 è ç 1 4 p ç kt èæèæ;tèèk L 2 ès 2 è; and thus, for t 0 2 ë0; 1è è19è è20è 1 t 0 kt èæèæ;tèèk L2 ès 2 èdt ç C èæèæ;t 0 èèè 1 2 : The ærst application of è19è is the calculation sup k^vèæ;tè, ^! 0 k L 2èS 2 è ç t2ët 0 ;1è = 1 t 0 1 t 0 æ L 2 ès 2 è dt kt èæèæ;tèèk L2 ès 2 èdt çc èæèæ;t 0 èèè 1 2 : The exponential decay èæèæ;t 0 èè then gives us the L 2 exponential convergence of Remark 2. In pursuit of Theorem 2, however, we are satisæed with the weaker statement è21è ^vèæ;tè! ^! 0 in L 2 ès 2 ; R 3 èast!1:

12 604 peter miles topping A consequence of this is that è22è ^vèæ;tè * ^! 0 weakly in W 1;2 ès 2 ; R 3 èast!1: This is because otherwise we could pick a sequence of times ft i g to give è23è jh^vèæ;t i è;æi,h^! 0 ;æij éæ; where æ is some test function, æé0, and hæ; æi is the inner product of W 1;2 ès 2 ; R 3 è. Then, as the total energy E is bounded èeèæèæ;tèè ç Eèæèæ; 0èèè we could pass to a subsequence of times èalso called ft i gè such that è24è ^vèæ;t i è *æweakly in W 1;2 ès 2 ; R 3 èasi!1; for some æ. The convergence would thus be strong in L 2 ès 2 ; R 3 è, and so by è21è we would have æ =^! 0. There would then be a contradiction between è23è and è24è. So è22è holds, which is part èiè of Theorem 2. Of course, as W 1;2 is compactly embedded in L p for any p 2 ë1; 1è, the convergence in è22è tells us that ^vèæ;tè! ^! 0 in L p ès 2 ; R 3 èast!1: We now proceed to consider the local oscillation of energy. r;sé0, deæne a cut-oæ function ' : R 2! R by 8 é 'èxè= 1+ é: 1 s 1 if jxj çr; èr,jxjè if réjxj ér+ s; 0 if jxj çr + s; For and deæne the `cut energy' æ w of a map w : R 2! R 3 by We also write æ w =æ èr;sè w = 1 2 R 2 ' 2 jrwj 2 : æètè =æ èr;sè ètè =æ èr;sè vèæ;tè : The cut energy is about the point in S 2 corresponding to the point 0 2 R 2, but this could be any point by taking a diæerent chart. The energy æ evolves according to dæ dt = ' 2 rv i :rt i =, ' 2 jt j 2 ç 2, 2 'èr':rv i èt i : R 2 R 2 R 2

13 rigidity in the harmonic map heat flow 605 Abandoning the ærst term on the right, and using Híolder's inequality, we estimate dæ dt ç 2 s æ 1 2 kt kl2 ès 2 è; where we are assuming that r + s é 1toavoid an extra constant. Integrating this, we see that the cut energy can only vary within the restriction æ ètè, æ 2 èt0 è ç 1 t kt èæèæ;çèèk L2 ès s 2 èdç; t 0 and thus, by è19è, è25è æ 1 2 ètè, æ 1 2 èt0 è ç C s 0 èèè 1 2 ; where tét 0, of course. The power of è25è is evident. It would be required to establish the exponential convergence of part èiiiè of Remark 2. However, we will be using the simple consequence that there exists a number l èr;sè such that è26è æ èr;sè ètè! l èr;sè as t!1: For any æçç S 2 nf^x 1 :::^x m g this provides us with the uniform control of concentration EèR; æè é" 1 ; for some R, enabling us to apply Lemma 3 to get k^vèæ;tèk C k èæè ç Cèkè uniformly for t 2 ë1; 1è; for all k. We can deduce the convergence of a subsequence of any sequence ^vèæ;t i èinc k èæè for any k, and hence establish part èiiè of Theorem 2 via the obvious contradiction argument. Having established part èiiè, part èiiiè of Theorem 2 then follows from a further application of è26è. q.e.d. Before proving Theorem 3, we need a perturbation result for ænite time intervals. Lemma 4. Suppose we have two solutions of the heat equation, which we write as maps ^v and ^w from S 2 æ ë0; 1è to S 2,! R 3, with initial maps ^v 0 and ^w 0 from S 2 to S 2,! R 3. Suppose moreover that T é 0 and that the æow ^v has no bubbles up to time t = T èin other words ^v is smooth for t 2 ë0;të.è

14 606 peter miles topping Then for all "é0, there exists æé0 independent of ^w such that if k^v 0, ^w 0 k W 1;2 ès 2 è éæ; then k^vèæ;tè, ^wèæ;tèk W 1;2 ès 2 è é"for all t 2 ë0;të: Proof. èlemma 4è. We sketch the proof, which essentially follows from the work of Struwe. For " 1 é 0wemaychoose R suæciently small so that sup E èx;2rè è^vèæ;tèè é " 1 èx;tè2s 2 æë0;t ë 4 : This is possible as ^v is regular for t 2 ë0;të. Then for k^v 0, ^w 0 k W 1;2 ès 2 è suæciently small, we may ensure that sup E èx;2rè è^w 0 è é " 1 x2s 2 : 2 Thus by Struwe's local control on the increase of energy ë6, Lemma 3.6ë, for çé0 suæciently small, we have that sup E èx;rè è^wèæ;tèè é" 1 : èx;tè2s 2 æë0;çë Here ç is dependent on^v only in terms of R, and essentially independent of ^w. The lemma then follows for T ç ç by ë6, Remark 3.9ë. By dividing up the interval ë0;tëinto intervals of length no more than ç and applying the lemma for T ç ç iteratively, we establish the lemma for general, ænite, T. q.e.d. Proof. ètheorem 3è. As in the proof of Theorem 2 we will assume without loss of generality that the body map and all the bubbles are anti-holomorphic, rather than holomorphic. The basic idea of the proof is to use Lemma 4 to show that the æows stay close until is small, and then use the techniques we developed in the proof of Theorem 2. We set " 0 as in Lemma 1 in anticipation. We will distinguish between the body maps of the æows ^v and ^w by calling them ^v 1 and ^w 1 respectively. Inkeeping with our previous notation, v and w will be the maps from R 2 æë0; 1è associated to ^v and ^w èthe maps from S 2 æ ë0; 1èè via stereographic projection.

15 rigidity in the harmonic map heat flow 607 Part èiè of Theorem 3 will follow from è20è. For any ç 1 ;ç 2 é 0, we may choose T suæciently large so that from part èiè of Theorem 2 and from è18è and k^vèæ;tè, ^v 1 k L 2 éç 1 for t ç T; è^vèæ;tèè é minèç 2 ; " 0 è for t ç T; 2 k^vèæ;tè, ^v 1 k W 1;2 èæè éç 3 for t ç T; from part èiiè of Theorem 2. Then for any ç 4 é 0, we may apply Lemma 4toændæé0 such that providing k^v 0, ^w 0 k W 1;2 ès 2 è éæ,wehave è27è k^vèæ;tè, ^wèæ;tèk W 1;2 ès 2 è é minèç 4 ; " 0 2 ;"è for all t 2 ë0;të. Therefore we must have è^wèæ;tèè ç è^wèæ;tèè é minèç 2 + ç 4 ;" 0 è for t ç T; and so we may use è20è to estimate k ^wèæ;tè, ^w 1 k L 2 écèç 2 + ç 4 è 1 2 for t ç T: Combining the above, we ænd that k^vèæ;tè, ^wèæ;tèk L 2 ç k^vèæ;tè, ^v 1 k L 2 + k^v 1, ^vèæ;tèk L 2 +k^vèæ;tè, ^wèæ;tèk L 2 +k ^wèæ;tè, ^w 1 k L 2 + k ^w 1, ^wèæ;tèk L 2 é 2ç 1 + ç 4 +2Cèç 2 + ç 4 è 1 2 ; for t ç T, and thus by taking ç 1 ;ç 2 ;ç 4 suæciently small and using è27è again, we establish part èiè of Theorem 3. To establish part èiiè we must control locally the oscillation of the ærst order part of the W 1;2 norm. By adapting the argument below, with w = v, we could establish the exponential convergence of part èiiè of Remark 2. With ' and æ as in the proof of Theorem 2 we calculate ç d 1 ç ' dt 2 R 2 jrèwèæ;tè, v 1 èj 2 è28è 2 ç = d dt æ wèæ;tè, d ' dt çr 2 rwèæ;tè:rv 1 : 2

16 608 peter miles topping The second term on the right-hand side is controlled by æ æ d dt ç çæ æææ æ ' 2 rwèæ;tè:rv 1 = æææ R æ ' 2 rt èwèæ;tèè:rv 1 2 R 2 ç Cèv 1 ;r;sèkt èwèæ;tèèk L 2èS 2 è; where we have integrated by parts and used Híolder's inequality. Together with è25è and è19è we nowhave enough information to integrate è28è giving 1 2 R 2 ' 2 jrèwèæ;tè, v 1 èj 2, 1 2 t R 2 ' 2 jrèwèæ;tè, v 1 èj 2 ç æ wèæ;tè, æ wèæ;t è + Cèv 1 ;r;sèkt èwèæ;çèèk L2 ès 2 èdç T ç ç ç æ 1 2 wèæ;tè +æ1 2 C wèæ;t è s Cèv1 èwèæ;tèèè 1 2 ç C èwèæ;tèèè 1 2 ç Cèç2 + ç 4 è; where the constant C on the ænal line is independent of the æow w assuming we insist that k^v 0, ^w 0 k W 1;2 ès 2 è é 1 èfor exampleè so that we have a bound on the energy of wèæ;tè. Taking ç 2 and ç 4 suæciently small, we may make the right-hand side as small as we desire. Consequently, for any ç 5 é 0we can ensure that k ^wèæ;tè, ^v 1 k W 1;2 èæè,k^wèæ;tè, ^v 1 k W 1;2 èæè ç ç 5 : Combining everything again, we see that k ^wèæ;tè, ^vèæ;tèk W 1;2 èæè çk ^wèæ;tè, ^v 1 k W 1;2 èæè + k^v 1, ^vèæ;tèk W 1;2 èæè çç 5 +èç 3 + ç 4 è+ç 3 ; for t ç T,soby taking ç 3 ;ç 5 suæciently small and making ç 4 smaller if necessary èand using è27è againè we establish part èiiè of Theorem 3. q.e.d. 5. An example of non-uniqueness As promised earlier, we now give a counter-example to part èiè of Theorem 2 when the condition that the target is S 2 is dropped. No bubbling occurs. The æow has a `winding' behaviour and has a circle of

17 rigidity in the harmonic map heat flow 609 accumulation points. The example will also show that a perturbation of a locally energy minimising harmonic map may move far away under the heat æow. It therefore contrasts with the work of Leon Simon èë5ëè in which these phenomena are ruled out under the hypothesis that the target is real analytic. Let the domain be S 2 and the target R 2 æ S 2. It is not important that R 2 is non-compact - as we shall see, we are only concerned with a bounded region, so we could change it to a æat 2-torus. We give the domain the standard metric, but give the target a warped metric - if g and h are the standard metrics on R 2 and S 2 respectively, then at a point èz; xè 2 R 2 æ S 2,we deæne the metric to be gèzè+fèzèhèxè where f : R 2! R + is to be determined. In other words, the target is R 2 æ f S 2. We consider initial maps of the form u 0 èxè =èz 0 ;xè where z 0 is independent ofx. Such maps give solutions of the heat equation è4è of the form uèx; tè =èzètè;xè, where z : R +! R 2, and z evolves according to è29è dz dt =,rfèzètèè: In other words, we have reduced the heat æow to ænite-dimensional gradient æow for z on the function f. It remains to choose the function f so that z may not have a unique limit, and so that moving z an arbitrarily small amount from a point z 0 with rfèz 0 è = 0 èwhich corresponds to a harmonic mapè will make the solution of è29è move away from z 0. To achieve this we take a `downwardly spiralling gramophone record' fèr;çè= è 1 if r ç 1; 1+e, 1 r,1 ç sinè 1 r,1 + çè+2 ç if ré1; where èr;çè are polar coordinates. Taking initial conditions with r = 2 say, the solution for z will spiral in to give the circle r = 1 as the accumulation set. Moreover, any point with r = 1 is a stationary point, but perturbing r to be slightly larger will make the solution of è29è spiral around, and move at least a distance 2 away. Remark 6. Of course, we are not restricted to using S 2 in the example above. In particular, the same idea will work for domains of any dimension.

18 610 peter miles topping 6. Acknowledgements The problem of uniqueness of the positions of bubbles was brought to the attention of the author by Leon Simon during a visit to Stanford University. The author would like to take this opportunity to thank Professors Schoen and Simon for their hospitality during this visit. The author would like to thank Professor James Eells for his encouragement of all the author's work on the harmonic map heat æow. The author is indebted to his supervisor Mario Micallef for, among other things, introducing him to the harmonic map heat æow, showing enthusiasm in this work, and encouraging the use of complex analytic techniques. This work will constitute part of the author's doctoral thesis ë7ë. Support has been given by the EPSRC, award number , and The Leverhulme Trust. References ë1ë W. Ding & G. Tian, Energy identity for a class of approximate harmonic maps from surfaces, Preprint, ë2ë J. Eells & J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 è1964è ë3ë A. Freire, Uniqueness for the harmonic map æow in two dimensions, To appear in Calculus of Variations & P.D.E. ë4ë J. Qing, On singularities of the heat æow for harmonic maps from surfaces into spheres, Comm. Anal. Geom. 3 è1995è ë5ë L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. 118 è1983è ë6ë M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Helv. 60 è1985è ë7ë P. Topping, The harmonic map heat æow from surfaces, PhD Thesis, University of Warwick, ë8ë J. C. Wood, Harmonic mappings between surfaces, PhD Thesis, University of Warwick, ë9ë W. P. iemer, Weakly diæerentiable functions, Graduate Texts in Math. 120, Springer, Berlin, University of Warwick, England