A rigidity estimate for maps from S 2 to S 2 via the harmonic map flow
aa r X i v : . [ m a t h . DG ] S e p A RIGIDITY ESTIMATE FOR MAPS FROM S TO S VIA THE HARMONIC MAP FLOW
Peter M. Topping22 September 2020
Abstract
We show how a rigidity estimate introduced in recent work of Bernand-Mantel,Muratov and Simon [1] can be derived from the harmonic map flow theory in [9].
For maps u ∈ W , ( S , S ), consider the harmonic map energy E ( u ) = 12 ˆ | Du | . Following Schoen-Uhlenbeck [4] we know that such a map can be approximated in W , ( S , S ) by smooth maps u i , whose degree will stabilise at some integer that givesa well-defined notion of degree of u . See also [7, Theorem 6.2]. For smooth maps w from a surface to any Riemannian manifold, a simple computation confirms that E ( w )is always at least the area of the image of w , with equality if w is weakly conformal.This implies that degree k ∈ N maps from S to itself must have energy at least 4 πk ,with equality implying that the map is a rational map. A special case of this is that if u is of degree one then E ( u ) ≥ π with equality if and only if u is a M¨obius map.The following theorem was recently proved by Bernand-Mantel, Muratov and Simon[1]. Theorem 1.1.
There exists a universal constant
C < ∞ such that for each u ∈ W , ( S , S ) of degree one, there exists a M¨obius map v : S → S such that ˆ | D ( u − v ) | ≤ C [ E ( u ) − π ] . The purpose of this note is to show that the estimate above can be derived from thetheory of the harmonic map flow developed in [9].1
The harmonic map flow from surfaces
The harmonic map flow [2] is the L -gradient flow for the harmonic map energy. If weview a flow u : S × [0 , T ) → S ֒ → R as taking values in R , then the equation in thiscase can be written ∂u∂t = τ ( u ) := (∆ u ) T where (∆ u ) T is the projection of ∆ u onto the tangent space of the target, i.e. (∆ u ) T =∆ u + u | Du | . The energy E ( t ) := E ( u ( t )) decays according to dEdt = −k τ ( u ) k L ( S ) . (2.1)In 1985 Struwe [6] initiated a theory for the harmonic map flow in the case that thedomain is a surface, as it is here. He showed how one can start a flow with smooth oreven W , initial data, giving a global weak solution that is smooth away from finitelymany points in space-time, at which bubbling occurs. At each finite-time singularity,concentrated energy at the singular points is thrown away by taking a weak limit, andthe flow restarted. Thus, the energy drops down at each singular time by at least theminimum energy of one bubble, i.e. of a nonconstant harmonic map from S to thetarget. However, the map t u ( t ) is continuous into L , even across singular times.(Later [10], different continuations through certain singularities were constructed thatdid not require a drop in energy, but we will not need them here.)Concerning the asymptotics at infinite time, there exists a sequence t i → ∞ such thatmaps u ( t i ) converge smoothly to a limiting harmonic map u ∞ away from finitely manybubble points. (See, for example, [7].) In general, even in the absence of bubbling,convergence of the form u ( t ) → u ∞ as t → ∞ may fail, even in L , see [8, 9]. However,a theory was developed in [9, 11] concerning such uniform convergence for maps from S to itself. A key lemma from that work, which will be useful to us now, gives arelationship between the tension field of a map u and its excess energy. It differedfrom previous estimates of ‘Lojasiewicz-Simon’ type [5] in that it could handle singularobjects. In particular the map u below is not asked to be W , close to a harmonicmap. Lemma 2.1 ([9, Lemma 1]) . There exist universal constants ǫ > and κ > suchthat if a smooth degree k ∈ Z map u : S → S satisfies E ( u ) − π | k | < ǫ , then E ( u ) − π | k | ≤ κ k τ ( u ) k L . As is well known, such an estimate gives control on the gradient flow. Indeed, given asmooth solution u : S × [0 , T ] → S that is of degree k ∈ Z , and satisfies E ( u ) − π | k | <ǫ at time t = 0 (and therefore also for later times) we can compute using (2.1) andthen Lemma 2.1 that − ddt [ E ( u ) − π | k | ] = 12 [ E ( u ) − π | k | ] − k τ ( u ) k L ≥ κ k τ ( u ) k L t = s ∈ [0 , T ) to t = T gives ˆ Ts k τ ( u ) k L dt ≤ κ (cid:16) [ E ( u ( s )) − π | k | ] − [ E ( u ( T )) − π | k | ] (cid:17) ≤ κ [ E ( u ( s )) − π | k | ] . (2.2)Since ∂u∂t = τ ( u ), the flow then cannot move far in L : k u ( T ) − u ( s ) k L ≤ κ [ E ( u ( s )) − π | k | ] . (2.3) Proof.
First, if E ( u ) = 4 π , then u must be a M¨obius map and we can choose v = u , sofrom now on we may assume that E ( u ) > π . By approximation, using the definitionof degree, it suffices to prove the result for u ∈ C ∞ ( S , S ). Since the estimate isinvariant under pre-composition by M¨obius maps, it suffices to prove the theorem for u equal to a map u with the property that ´ S u = 0 ∈ R . That this can be achievedfollows from a topological argument: For a ∈ B , let ϕ a : S → S be the M¨obiusmap that fixes ± a | a | , whose differential does not rotate the tangent spaces at ± a | a | , andwhich when extended to a conformal map B B will send the origin to a . Thusas a approaches some a ∈ S = ∂B , the composition u ◦ ϕ a converges to u ( a ) awayfrom − a , so π ´ u ◦ ϕ a → u ( a ). Thus the map a π ˆ u ◦ ϕ a extends to a continuous map Φ from B to itself that agrees with the degree one map u on the boundary S . A topological argument then tells us that Φ is surjective, sinceotherwise Φ could be homotoped to a continuous map ˜Φ from B to S that restrictsto u : S → S . But then ˜Φ would provide a homotopy from the degree one map u to a constant map, a contradiction. In particular, there exists a ∈ B such thatΦ( a ) = 0 ∈ B . We can then set u := u ◦ ϕ a to achieve our objective. For a relatedargument in which one post -composes with M¨obius maps see Li-Yau [3].Let now ǫ be as in the key lemma 2.1. For later use, if necessary we reduce ǫ > κ ) ǫ ≤ π. (3.1)We may assume that our map u satisfies E ( u ) − π < ǫ since otherwise the theoremis vacuously true.To prove the theorem, run the harmonic map flow starting with u . By (2.3), the flowis constrained in how far it can move in L . Because of the balancing ´ S u = 0, thisimplies that ´ S u ( t ) remains close to the origin, which precludes bubbling both at finiteand infinite time. More precisely, because ddt ˆ u = ˆ τ ( u )3e have (cid:12)(cid:12)(cid:12)(cid:12) ddt ˆ u (cid:12)(cid:12)(cid:12)(cid:12) ≤ √ π k τ ( u ) k L , and therefore, integrating from 0 to t using (2.2) we have (cid:12)(cid:12)(cid:12)(cid:12) ˆ − u ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ κ √ π [ E ( u ) − π ] . (3.2)Suppose we develop a singularity at a finite or infinite time T ∈ (0 , ∞ ]. If we pick t i ↑ T such that u ( t i ) ⇀ w weakly in W , , then we must have E ( w ) ≤ E ( u ) − π ,since the singularity loses at least 4 π of energy (that being the least possible energy ofa nonconstant harmonic map from S to itself) but also because ˆ − u ( t i ) → w := ˆ − w we have | w | ≤ κ √ π [ E ( u ) − π ] . But the Poincar´e inequality tells us that ˆ S | w − w | ≤ E ( w ) , and so integrating the inequality1 = | w | ≤ | w − w | + 2 | w | , we obtain 4 π ≤ ˆ | w − w | + 8 π | w | ≤ E ( w ) + 8 κ [ E ( u ) − π ] ≤ (2 + 8 κ ) [ E ( u ) − π ] ≤ π, (3.3)by (3.1), giving a contradiction.We deduce that the flow exists for all time and converges smoothly to a M¨obius map v .Here we only need convergence at some sequence of times t i → ∞ , although we haveconvergence as t → ∞ by [9].We can also pass the inequality (3.2) to the limit t → ∞ to give | v | ≤ κ √ π [ E ( u ) − π ] ≤ κ (cid:16) ǫ π (cid:17) ≤ , by (3.1). This estimate prevents v from being too concentrated, and we can deducethat | Dv | ≤ c , (3.4)for some universal c . This follows by noticing that every M¨obius map can be writtenas one of the maps ϕ a followed by a rotation of S . Another way of making this precise4s by arguing by contradiction: If such an estimate (3.4) were not true, then we wouldtake a sequence of M¨obius maps v i with | v i | ≤ but with sup | Dv i | → ∞ . After abubbling analysis (passing to a subsequence) the maps v i would converge weakly in W , to a constant map v ∞ : S → S with | v ∞ | ≤ , a contradiction.Returning to (2.3), we find that k u − v k L ≤ κ [ E ( u ) − π ]. But we can also compute ˆ | D ( u − v ) | = ˆ | Du | + ˆ | Dv | − ˆ h Du , Dv i and because − ∆ v = v | Dv | , we can handle the final term using − ˆ h Du , Dv i = − ˆ u ( − ∆ v ) = − ˆ u v | Dv | = ˆ | u − v | | Dv | − ˆ | Dv | , (3.5)where we have used that | u | = | v | = 1. Combining, we obtain ˆ | D ( u − v ) | = ˆ | Du | − ˆ | Dv | + ˆ | u − v | | Dv | ≤ E ( u ) − π ] + c k u − v k L ≤ C [ E ( u ) − π ] , (3.6)for universal C . Remark 3.1.
At the start of the argument we balanced our map to have ‘centreof mass’ at the origin. Without this step we could expect the harmonic map flowto generate a finite-time singularity. Indeed in [8, Theorem 5.5] we showed that forarbitrarily small ǫ >
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