Harmonic Map Flow with Low $\Bar {\Partial} $-Energy

$Harmonic Map Flow with Low $\Bar {\Partial} $-Energy$

HARMONIC MAP FLOW FOR ALMOST-HOLOMORPHIC MAPS

CHONG SONG AND ALEX WALDRON

Abstract. Let Σ be a compact oriented surface and N a compact K¨ahlermanifold with nonnegative holomorphic bisectional curvature. For harmonic map ﬂow starting from an almost-holomorphic map Σ → N (in the energy sense), the limit at each singular time extends continuously over the bubble points and no necks appear.

Contents 1. Introduction 1 2. Basic identities 3 3. Epsilon-regularity 10 4. Estimate on the curvature scale 16 5. Decay estimate in annular regions 19 6. H¨older continuity and bubble-tree convergence 25 References 31

1. Introduction 1.1. Background. Let M, g and N, h be compact Riemannian manifolds. For any C1 map u M N, we may define the Dirichlet energy ( ) ( ) 1 2 ∶ → E u du dVg. 2 M Critical points of E u are referred( to) as= harmonicS S S maps and play a fundamental role in geometric analysis. Harmonic map flow( )is the negative gradient flow of the Dirichlet functional: ∂u arXiv:2009.07242v2 [math.DG] 20 Sep 2021 (1.1) tr du. ∂t g In 1964, Eells and Sampson [7] introduced= the∇ evolution equation (1.1) and proved their foundational theorem: if N has nonpositive sectional curvature, then harmonic map flow smoothly deforms any initial map to a harmonic map in the same homotopy class. The situation becomes more challenging when the target manifold is allowed to have positive sectional curvatures. It is common to assume that the domain has dimension two, where the Dirichlet functional is conformally invariant and enjoys a rich variational theory [20]. In 1985, Struwe [25] constructed a global weak solution of (1.1) with arbitrary initial data in W 1,2. Struwe’s solution is regular away from a finite set of spacetime points where the Dirichlet energy may possibly concentrate, forming a singularity. In 1992, Chang, Ding, and 2 CHONG SONG AND ALEX WALDRON

Ye [4] proved that these finite-time singularities must occur in the scenario of rotationally symmetric maps between 2-spheres; they are an inevitable feature of the theory.1 During the 1990s, a detailed picture of singularity formation in 2D harmonic map flow emerged from the work of several authors. Given a singular time 0 T , the limit 1,2 u T limt→T u t , referred to as the body map, exists weakly in W and smoothly away from the singular set (this already follows from [25]). Based on a rescaling< < procedure∞ near 2 each( ) singular= point( ) xi,T , one obtains a bubble tree consisting of finitely many harmonic 2 maps φi,j S N satisfying the energy identity [15, 6, 33]: ( ) lim E u t E u T E φi,j . ∶ → t↗T i,j Moreover, necks cannot form between( ( )) the= bubbles( ( )) + atQ a given( ) point; i.e., for each i, the set 2 jφi,j S N must be connected [17, 9]. For some time, it was believed that the body map u T should always extend continuously ∪across( the) ⊂ singular set [9, 16]—that is, until 2004, when Topping [29] was able to construct an example in which u T has an essential singularity.( This) may have caused interest in the subject to decline in the following years. However, Topping’s( counterexample) depends on the construction of a pathological metric on the target manifold. In the same paper [29], he conjectured that when the metric on N is sufficiently well behaved (specifically, real-analytic), u T will have only removable discontinuities. The conjecture is usually taken to include the statement that necks cannot form between the bubbles and the body map, which is currently( ) known only at infinite time. Topping [30] proved that both properties follow from Höldercontinuity of the Dirichlet energy E u t with respect to t T, although this assumption is difficult to verify in practice.

1.2.( (Main)) results. This< paper establishes the continuity and no-neck properties under a set of hypotheses familiar from the classical theory of harmonic maps [24, 34] as well as later work on harmonic map flow. In the Kählersetting, the Dirichlet energy decomposes into holomorphic and anti-holomorphic parts: 2 ¯ 2 E u ∂u dVg ∂u dVg E∂ u E∂¯ u . Topping’s earlier work [28] on rigidity of (1.1) at infinite time was based on assuming that ( ) = S S S + S S S =∶ ( ) + ( ) u is almost (anti-)holomorphic, i.e., that either E∂ u or E∂¯ u is small. Liu and Yang [10] generalized Topping’s rigidity theorem using a Bochner technique that also requires a positivity assumption on the curvature of N. We make( ) the same( ) assumptions in our main theorem, which follows. Theorem 1.1. Given a compact Kählermanifold N, h with nonnegative holomorphic bi-

sectional curvature, there exists a constant ε0 0 as follows. Let Σ, g be a compact, oriented, Riemannian( surface) and u Σ 0, N a weak > solution (in Struwe’s sense) of harmonic map ﬂow with either E∂¯ u 0 ε0 or E∂ u 0 α ε0. For( each) singular time T , the map u T is C for each α∶ 1.×Moreover,[ ∞) → given any ( ( )) < ( ( )) < 1 The construction of [4] can easily< ∞ be adapted to( produce) ﬁnite-time blowup< starting from an almost- holomorphic initial map S2 → S2, in the sense of Theorem 1.1. 2See Parker [14] for an introduction to the bubble-tree concept. HARMONIC MAP FLOW FOR ALMOST-HOLOMORPHIC MAPS 3

sequence of times tn T or , the maps u tn sub-converge in the bubble-tree sense, without necks, and satisfy the energy identity. ↗ ∞ ( ) Note that by the proof of the generalized Frankel conjecture due to Mori [12], Siu-Yau [24], and Mok [11], our curvature assumption implies that N is biholomorphic to a Hermitian symmetric space, and to CPn in the strictly positive case. However, the metric h need not be symmetric: for example, any metric on the 2-sphere with nonnegative curvature satisﬁes 2 the hypothesis. Meanwhile, even in the case Σ N Sround, our results are new. We also note that the bubble-tree statement in Theorem 1.1 allows for an arbitrary sequence tn T or , by contrast with the work= discussed= above. This is possible because our results follow from parabolic estimates rather than an analysis of sequences of low-tension maps. ↗ ∞

1.3. Outline of proof. The proof of Theorem 1.1 involves the following three steps. First, we prove that ∂u¯ remains uniformly bounded along the ﬂow in our scenario (The- orem 3.5). This follows from a version of the standard ε-regularity estimate compatible with the split Bochner formulaS S (Proposition 3.3). Next, we show that a uniform Lq bound on the stress-energy tensor (2.5) implies the following estimate on the (outer) energy scale at a ﬁnite-time singularity:

q λ t O T t 2 t T .

1 For q 1, this strengthens the standard( ) = ( “Type-II”− ) ( ↗ bound) λ t o T t 2 for (1.1) in dimension two. The improvement is crucial; indeed, it fails in Topping’s counterexample (see [29],> Theorem 1.14e). In the setting of Theorem 1.1, the uniform( ) = bound( − ) on ∂u¯ implies an L2 bound on the stress-energy tensor, yielding the improved blowup rate with q 2 (Corollary 4.6).3 S S Last, we use this small amount of “spare time” before the blowup to construct a superso-= lution for the angular component of du in the neck region (§5.1-5.2). Using the stress-energy bound once more, we obtain a strong decay estimate on the full energy density (Theorems 5.5-5.6), leading directly to our main theorems (§6).

1.4. Acknowledgements. C. Song is partially supported by NSFC no. 11971400. A. Waldron is partially supported by DMS-2004661. The authors thank K. Gimre for editorial comments.

2. Basic identities In this section, we derive all of the identites required to obtain estimates along the ﬂow. The formulae in §2.1-2.3 apply to general domains and targets, while those in §2.4-2.6 spe- cialize to a surface domain and K¨ahlertarget.

3The q = 2 bound is well known in the rotationally symmetric case—see Remark 4.7 below. 4 CHONG SONG AND ALEX WALDRON

2.1. Intrinsic viewpoint on harmonic map ﬂow. Let M, g and N, h be compact Riemannian manifolds. Given a smooth map u M N, we write du for its diﬀerential, which may be viewed as a section of ( ) ( ) ∶ → u∗TN T ∗M. We shall use the notation , for the pullback of h to u∗TN, combined appropriately with g E = ⊗ on tensor products with T ∗M. The energy density of u is given by ⟨ ⟩ 1 1 e u du 2 gij ∂ u, ∂ u , 2 2 i j and the Dirichlet functional by ( ) = S S = ⟨ ⟩

E u Eg u e u dVg. M We denote by the pullback to u∗TN of the Levi-Civita connection of h, which we will ( ) = ( ) = S ( ) combine with that of g on tensors. Under an inﬁnitesimal∇ variation u u δu g g δg, → + we have → + 1 1 (2.1) δE u 2gij ∂ u, δu δgij ∂ u, ∂ u gij ∂ u, ∂ u gk`δg dV g 2 i j i j i j 2 k` g ij 1 ik lj 1 2 ij (2.2) ( ) = S (g ⟨ j∂iu,∇ δu ⟩ + g⟨ g ∂ku,⟩ + ∂lu ⟨ du g⟩ δgij dV) g 2 2 1 (2.3) = S − ⟨∇u , δu ⟩ +S, δg− dV ⟨. ⟩ + S S 2 g Here = − S ⟨T ( ) ⟩ + ⟨ ⟩

(2.4) u trg du is the tension field, and T ( ) = ∇ 1 (2.5) S u du du du 2g 2 2 is the stress-energy tensor. The former( ) = is⟨ a section⊗ ⟩ − ofS u∗STN, and the latter of Sym T ∗M. Recall that harmonic map flow is the evolution equation ∂u u ∂t for the map u u x, t . This is the negative= gradientT ( ) flow of the Dirichlet functional, where the metric g is fixed in time. Then (2.1) immediately gives the global energy identity

= ( ) t2 2 (2.6) E u t2 u dVg dt E u t1 t1 M for a suﬃciently regular solution( ( )) of+ S (1.1).S ST ( )S = ( ( )) Remark 2.1. In what follows, we shall always assume that u is a classical solution of harmonic map ﬂow on 0,T , where 0 T is included in the data of u. In particular, T need not be the maximal smooth existence time. [ ) < ≤ ∞ HARMONIC MAP FLOW FOR ALMOST-HOLOMORPHIC MAPS 5

2.2. Pointwise energy identity. The following brief calculation yields a useful “pointwise” version of (2.6). Using normal coordinates at a point, we take the divergence of S

1 2 div S iSij i∂iu, ∂ju ∂iu, i∂ju j du ∶ j 2 1 (2.7) ( ) = ∇ = ⟨∇ u , ∂ u ⟩ + ⟨∂ u, ∇ ∂ u ⟩ − ∇ Sdu S2 j i j i 2 j = ⟨T (u), ∂ju⟩ .+ ⟨ ∇ ⟩ − ∇ S S Here we have used the fact = ⟨T ( ) ⟩ (2.8) i∂ju j∂iu, which follows from torsion-freeness of the Levi-Civita connection(s). The identity (2.7) ∇ = ∇ appears as (2.10) in the paper of Baird and Eells [2]. Taking another divergence, we obtain div div S u , du u 2. On the other hand, for a solution u x, t of harmonic map flow, we have ( ( )) = ⟨∇T ( ) ⟩ + ST ( )S ∂e u ∂u ( ) , du u , du . ∂t ∂t ( ) We obtain the pointwise energy identity= c∇: h = ⟨∇T ( ) ⟩ ∂e u (2.9) u 2 div div S . ∂t ( ) + ST ( )S = ( ( )) 2.3. Bochner formula. Let (as above) be the connection induced on u∗TN T ∗M by the pullback of the Levi-Civita connection on N, coupled with that of M. The connection defines a (crude) Laplace operator∇ ∆ ∗ on . E = ⊗ Fixing p M, we let xi be normal coordinates at p, and ya be normal coordinates at u∇ p N. We shall denote the tensor components= −∇ ∇ ofEdu in these coordinates by ∈ { } { } ∂ ya u ( ) ∈ ua . i ∂xi ( ○ ) We then have = a a a ∆du j i iuj i jui ua , ua ( ) = ∇ ∇ = ∇j∇i i i j i u a MR k ua u∗NR a ub (2.10) = ∇j∇ + [∇ ∇ij ]i k ij b i u a MR k ua NR a ucudub = ∇jT ( ) − ij i k + ( cd b) i j i u a MRic ua NR ubucud. = ∇jT ( ) − jk k+ abcd i j i Here we have used (2.8) in the first line, and the Bianchi identities in the last line. = ∇ T ( ) + −∇ Now, given a map u M 0,T N, we shall write ∂t for the time-component of the covariant derivative induced by the pullback of the Levi-Civita connection to M 0,T . The identity ∶ × [ ) → du ∂u × [ ) ∂t ∂t ∇( ) = ∇ 6 CHONG SONG AND ALEX WALDRON

is obvious in normal coordinates. Assuming that u x, t is a solution of harmonic map ﬂow, we obtain du ( ) (2.11) u . ∂t ∇( ) Then (2.10) becomes = ∇T ( ) (2.12) ∆ ua gk` MRic ua gk` NR ub ucud. ∂t j jk ` abcd k j ` ∇ Taking an inner product − with du,= −and using the identity+ ∆e u ∆du, du du 2, we obtain ( ) = ⟨ ⟩ + S∇ S ∂ (2.13) ∆ e u du 2 h MRicij ua, ub gikgj` NR ua, ub, uc , ud . ∂t ab i j abcd i j k ` This yields the− diﬀerential ( ) = − inequalityS∇ S − + ∂ (2.14) ∆ e u du 2 Ae2 u Be u . ∂t Here, A is a constant times an− upper ( ) bound≤ −S∇ forS + the sectional( ) + curvature( ) of N, and B is a constant times a lower bound for the Ricci curvature of M. Remark 2.2. If the sectional curvature of N is non-positive, then one may let A− 0 in (2.14), which gives a uniform bound on e u by Moser’s Harnack inequality. This is the key point in the proof of Eells-Sampson’s Theorem. = ( )

2.4. Holomorphic splitting in the Kählercase. We now restrict to the case that M Σ is an oriented surface and N is a Hermitian manifold of complex dimension n. The complex- ified tangent spaces decompose as = (2.15) T ΣC T 1,0Σ T 0,1Σ,TN C T 1,0N T 0,1N. As an element of C , the differential of u decomposes under the splittings (2.15) as C= ⊕ = ⊕ ∂u ∂u¯ (2.16) E = E ⊗ du . ∂u¯ ∂u ⎛ ⎞ Here we abuse notation slightly; letting z= be a local conformal coordinate on Σ near a point ⎝ ⎠ α n p, and w α=1 local holomorphic coordinates on N near u p , we have ∂wα ∂ ∂wα ∂ { } ∂u dz , ∂u¯ (dz¯) ∂z ∂wα ∂z¯ ∂wα (2.17) ∂wα ∂ ∂wα ∂ ∂u¯ = dz ⊗ , ∂u = dz¯ ⊗ , ∂z ∂w¯α ∂z¯ ∂w¯α where = ⊗ = ⊗ ∂ 1 ∂ ∂ dz dx idy, i ∂z 2 ∂x ∂y ∂ 1 ∂ ∂ dz¯ = dx + idy, = − i , etc. ∂z¯ 2 ∂x ∂y = − = + HARMONIC MAP FLOW FOR ALMOST-HOLOMORPHIC MAPS 7

We shall also write ∂wα ∂ ∂wα ∂ w , w z ∂z ∂wα z¯ ∂z¯ ∂wα w¯z¯ = wz, w¯z wz¯.= Let h , denote the complex-linear extension of the metric to TN C. Then = = ∂ ∂ ∂ ∂ (2.18)( ) h , 0 h , , ∂wα ∂wβ ∂w¯α ∂w¯β and the complex n n matrix = = ∂ ∂ h ¯ h , × αβ ∂wα ∂w¯β

is Hermitian, with hαβ¯ hβα¯, and positive-definite.= Denote the Hermitian inner product corresponding to h by = (2.19) v, w h v, w¯ for v, w TN C. Denote the corresponding norm by , which agrees with the ordinary norm ⟨ ⟩ = ( ) on real tangent vectors. Suppose∈ now that N is Kähler.The Kähler formS ⋅ onS N can be written α β ωN ihαβ¯dw dw¯ . We also extend g complex-linearly, and let = ∧ ∂ ∂ (2.20) σ2 z g , 0. ∂z ∂z¯ The metric and Kählerform on Σ can( ) be= written > 2 2 g gΣ σ dz dz¯ dz¯ dz , ωΣ iσ dz dz.¯ Let = = ( ⊗ + ⊗ ) = ∧ 2 −2 2 −2 α β e∂ u ∂u σ wz σ hαβ¯w w¯z¯ (2.21) z 2 −2 2 −2 α β e ¯ u ∂u¯ σ w σ h ¯w w¯ . ∂( ) = S S = S z¯S = αβ z¯ z Then according to (2.16), the energy density decomposes as ( ) = S S = S S = 1 2 e u du e u e ¯ u . 2 ∂ ∂ On the other hand, the Kählerform( ) = onSN Spulls= back( ) + to ( ) ∗ ¯ u ωN ih ∂u ∂u ∂u¯ ∂u i w 2 w 2 dz dz¯ = z + z¯ ∧ + e u e ¯ u ω . = ∂S S − S ∂ S Σ∧ Thus = ( ( ) − ( )) ∗ (2.22) E∂ u E∂¯ u u ωN κ Σ is an invariant of the homotopy class of u; the energy of u satisfies ( ) − ( ) = S =∶

(2.23) E u E∂ u E∂¯ u 2E∂ u κ 2E∂¯ u κ.

( ) = ( ) + ( ) = ( ) − = ( ) + 8 CHONG SONG AND ALEX WALDRON

Hence, an (anti-)holomorphic map minimizes the Dirichlet energy within its homotopy class.

It also follows from (2.23) that both E∂ u t and E∂¯ u t decrease in time along a solution of harmonic map ﬂow. ( ( )) ( ( ))

2.5. Hopf differential and stress-energy tensor. We now define the Hopf differential

2,0 2 1,0 Φ u h du du Sym ΩM .

In local coordinates, we have ( ) = ( ⊗ ) ∈ Φ u h ∂u ∂u¯ ∂u ∂u¯

(2.24) h wz, wz h w¯z, w¯z h wz, w¯z h w¯z, wz dz dz ( ) = (( + ) ⊗ ( + )) 2 w , w dz dz, = ( ( z z¯ ) + ( ) + ( ) + ( )) ⊗ where we have used= (2.18).⟨ It⟩ follows⊗ from (2.24) that the Hopf diﬀerential of a harmonic map is holomorphic. We shall not use this fact, but only the following identity which holds for general maps.

Lemma 2.3. For a diﬀerentiable map u Σ N, the stress-energy tensor is given by

(2.25) S u∶ 2→ Re Φ u . In particular, the pointwise bound ( ) = ( )

(2.26) S u g 4 e∂ u e∂¯ u » holds. S ( )S ≤ ( ) ( ) Proof. Recall the deﬁnition (2.5) of S u . Since du is real, and in view of (2.18), we have ¯ du du h du du h ∂u (∂u) h ∂u ∂u¯ h ∂u ∂u h ∂u ∂u h ∂u¯ ∂u h ∂u¯ ∂u¯ h ∂u¯ ∂u h ∂u¯ ∂u¯ ⟨ ⊗ ⟩ = ( ⊗ ) = ⊗ + ( ⊗ ) + ⊗ + ⊗ w 2 w 2 dz dz¯ dz¯ dz + z ⊗ z¯ + ⊗ + ( ⊗ ) + ⊗ 2 w , w dz dz w , w dz¯ dz¯ = (S S +zS z¯S )( ⊗ + z¯ ⊗ z ) 1 2 + du(⟨ g 4 Re⟩ ⊗wz, w+z¯ ⟨dz dz⟩ . ⊗ ) 2 Rearranging yields (2.25). Applying= S S (2.24)+ and(⟨ Cauchy-Schwarz⟩ ⊗ ) to (2.25) yields (2.26). Remark 2.4. There is a strong analogy between harmonic maps from a Riemann surface to an (almost-)complex manifold and Yang-Mills connections on 4-manifolds. Letting j and J be the complex structures on Σ and N, respectively, deﬁne an operator

φ J φ j. ⋆ ∶ E → E 2 Then 1, and the diﬀerential du decomposes↦ ○ along○ the -eigenspaces of as + − ∗ = du du du ± ⋆ = + HARMONIC MAP FLOW FOR ALMOST-HOLOMORPHIC MAPS 9

where 1 du+ du J du j 2 Re ∂u,¯ 2 1 du− = (du + J ○ du ○ j) = 2 Re ∂u. 2 The stress energy tensor then takes= the( form− ○ ○ ) =

S 2 du+ du− ,

which can be compared with [32, Remark= ⟨ 2.8]. ⊗ ⟩

2.6. Split Bochner formula. We also require a split Bochner formula that goes back to Schoen and Yau [22], Toledo [27], and Wood [34] for harmonic maps, and has been exploited in a parabolic context by Liu and Yang [10]. Since Σ and N are K¨ahler,their complex structures commute with the C-linear extension of to C. Hence, the operators on the LHS of (2.12) preserve the splittings (2.15). Taking an inner product with ∂u¯ in (2.12), we obtain ∇ E 1 ∂ (2.27) ∆ e ¯ u ∂u¯ ∂u,¯ I ∂u,¯ II , 2 ∂t ∂ where I and II are the terms− on the( ) RHS= −S∇ of (2.12).S + a Thesef + a mayf be expanded as follows.

Since RicΣ KΣg, we have

¯ ¯ 2 (2.28) = ∂u, I KΣ ∂u KΣe∂¯ u .

∂ ∂ For the second term, choose aa conformalf = − coordinateS S = − z (x ) iy such that ∂x , ∂y form an orthonormal basis at p (in particular, σ2 p 1 ). At the point p, we have 2 = + { } ∂ ∂ ∂ ∂ ∂u,¯ II NR ∂u¯ (, du) = , du , du ∂x ∂y ∂x ∂y (2.29) ∂ ∂ ∂ ∂ a f = NR ∂u¯ , du , du , du . ∂y ∂x ∂y ∂x Since + ∂ ∂ ∂ ∂ ∂ ∂ , i ∂x ∂z ∂z¯ ∂y ∂z ∂z¯ we obtain = + = − ∂ ∂ ∂ du wα wα w¯α w¯α ∂x z z¯ ∂wα z z¯ ∂w¯α ∂ ∂ ∂ du = (i wα+ wα) + (i w¯+α w¯)α ∂y z z¯ ∂wα z z¯ ∂w¯α ∂ ∂ ∂ ∂ ∂u¯ = w(α − , ) ∂u¯ + ( −iwα) . ∂x z¯ ∂wα ∂y z¯ ∂wα = = − 10 CHONG SONG AND ALEX WALDRON

Substituting into (2.29), and using the fact that NR is a section of Λ1,1 Λ1,1, we obtain ¯ N α β β γ γ δ δ ∂u, II Rαβγ¯ δ¯wz¯ i w¯z w¯z¯ wz wz¯ i w¯z w¯z¯ ⊗ N α β β δ δ γ γ R ¯¯ w i w¯ w¯ w¯ w¯ i w w a f = αβδγ z¯ z − z¯ ( z + z¯ ) z − z¯ N α β β γ γ δ δ (2.30) R ¯ ¯ iw w¯ w¯ i w w w¯ w¯ + αβγδ z¯ − z z¯ + z (z¯ − z ) z¯ N α β β δ δ γ γ R ¯¯ iw w¯ w¯ i w¯ w¯ w¯ w¯ + αβδγ(− z¯ ) z + z¯ ( z − z¯ ) z + z¯ N α β γ γ δ δ γ γ δ δ 2 R ¯ ¯w w¯ w w w¯ w¯ w w w¯ w¯ , + αβγ(δ− z¯ )z +z z¯ −z z¯( + z ) z¯ z z¯ where we have combined the ﬁrst with the fourth, and the second with the third lines. This = − ( + ) − + ( − ) + yields ¯ N α β γ δ γ δ ∂u, II 4 Rαβγ¯ δ¯wz¯ w¯z wz w¯z¯ wz¯w¯z (2.31) −4 N α β γ δ α β γ δ σ R ¯ ¯ w w¯ w w¯ w w¯ w w¯ q u q u . a f = αβγδ z¯ z z −z¯ z¯ z z¯ z 1 2 Returning to (2.27), we obtain = − =∶ ( ) + ( ) 1 ∂ ∆ e ¯ u ∂u¯ K e ¯ u q u q u . 2 ∂t ∂ Σ ∂ 1 2 The condition that h have− nonnegative ( ) = −S∇ holomorphicS − ( bisectional) + ( ) + curvature( ) (see Goldberg and Kobayashi [8] or Schoen and Yau [24], Ch. VIII) is equivalent to the assumption that

q1 u 0. It is also clear that 2 q2 u CN e∂¯ u , ( ) ≤ where C is a constant times the supremum of the holomorphic sectional curvature of N. N ( ) ≤ ( ) Similar formulae hold for the holomorphic part of the energy density, e∂ u . We have shown: ( ) Lemma 2.5 (Liu and Yang [10]). Assume that N has nonnegative holomorphic bisectional curvature. For a solution u x, t of (HM), we have 1 ∂ (2.32) (∆ )e u ∂u 2 K e u C e u 2 2 ∂t ∂ Σ ∂ N ∂

1 ∂ 2 2 (2.33) − ∆ e ¯(u) ≤ −S∇∂u¯ S − K e ¯(u) + C e ¯(u) . 2 ∂t ∂ Σ ∂ N ∂ − ( ) ≤ −S∇ S − ( ) + ( )

3. Epsilon-regularity 3.1. Scalar heat inequality. In this subsection, we estimate a weak solution v x, t of the diﬀerential inequality ∂ ( ) (3.1) ∆ v Av3 Bv ∂t g on a ball inside the compact surface Σ−, g .4These≤ estimates+ will lead to ε-regularity results of the form we require. ( ) 4Since our results are local, the statements also apply to complete Σ where g has “bounded geometry,” i.e., uniform bounds on each derivative of the Riemann tensor and injectivity radius bounded from below. HARMONIC MAP FLOW FOR ALMOST-HOLOMORPHIC MAPS 11

We shall let ε0 0 be a constant depending only on A, which may decrease with each subsequent appearance. The constant R0 0 will depend on B and the geometry of Σ, and may also decrease.> We shall always assume > (3.2) 0 R R0

where R0 is small enough that any relevant Sobolev< < inequality holds on BR p with a constant independent of R and p, and abbreviate BR BR p . For 0 r R, denote the annulus ( ) R ¯ Ur B= R (Br). < <

Let ϕ 0 be a smooth cutoﬀ function, with= supp∖ϕ BR, ϕ 1 on BR~2, and ϕ L∞ 4 R.

1 Lemma≥ 3.1 (Struwe [26], Lemma III.6.7). Given any⊂ function≡ v on BR of SobolevY∇ Y class< H~ , we have

(3.3) v2ϕ2 C R2 v 2ϕ2 v2 B U R R R~2 S ≤ S S∇ S + S (3.4) v4ϕ2 C v2 v 2ϕ2 R−2 v2 . B B U R R R R~2 Proof. The estimateS (3.3) follows≤ fromS the Poincar´einequalityS S∇ S + S and Young’s inequality:

v2ϕ2 CR2 vϕ 2

2 2 2 2 2 S ≤ CR S S∇ (v ϕ)S 2ϕv v, ϕ v ϕ

2 2 2 2 2 ≤ CR S S∇vS ϕ + v ⟨ϕ∇ ∇ ⟩ + S∇ S

≤ C RS2 S∇ vS 2 + S∇v2 S . B U R R R~2 For (3.4), we refer to Lemma≤ III.6.7 S of [26].S∇ S The+ S proof there gives an inequality

v4ϕ2 C v2 v 2 ϕ2 v2 ϕ 2 . supp ϕ R ≤ S∇ S + S∇ S Since supp ϕ UR~2, Sthe estimate (3.4)S follows.S Proposition∇ 3.2⊂ (Cf. [32], Proposition 3.4). Let v be a nonnegative continuous weak solution of (3.1) on BR 0,T . Suppose that

2 2 × [ ) v 0 ε0, sup v t ε ε0. B U R R 0≤t

≤ < > 12 CHONG SONG AND ALEX WALDRON

Proof. Let ε1 ε0, and deﬁne 0 T1 T to be the maximal time such that

2 (3.6) > < v t≤ ε1 0 t T1 . BR Let ϕ be as above. MultiplyingS (3.1)( ) by< ϕ2v and(∀ integrating≤ < ) by parts, we obtain 1 d ϕ2v2 ϕ2v v A ϕ2v4 B ϕ2v2. 2 dt Applying Young’s inequality,S we + obtainS ∇( ) ⋅ ∇ ≤ S + S 1 d 1 ϕ2v2 ϕ2 v 2 2 ϕ 2v2 A ϕ2v4 B ϕ2v2. 2 dt 2

For 0 t T1, weS apply (3.4) + onS theS∇ RHSS ≤ andS rearrange,S∇ S + toS obtain + S 1 d 1 ≤ < ϕv 2 CAε ϕ2 v 2 C 1 Aε R−2ε B ϕv 2 . 2 dt 2 1 1 Applying (3.3)S on( the) left-hand+ − side, andS rearrangingS∇ S ≤ ( again,+ we) obtain+ S ( ) d α (3.7) ϕv 2 ϕv 2 C 1 Aε R−2ε, dt R2 1 where S ( ) + S ( ) ≤ ( + ) 1 1 α CAε CBR2. C 2 1 −1 −1 −1 We may assume that ε1 4CA =and R0− 4CB − , so that α 8C . Rewrite (3.7) as < ( ) < ( ) > ( ) d 2 2 eαt~R ϕv 2 CR−2εeαt~R dt and integrate in time, to obtain S ( ) ≤ v t 2 e−αt~R2 v 0 2 Cε 1 e−αt~R2 . BR~2 BR ( ) ≤ ( ) + − This establishes (3.5)S for 0 t T1. S Assume, for the sake of contradiction, that T1 T. For t T1, (3.5) reads ≤ < 2 v ε0< Cε. < BR~2 Provided that S ≤ +

2 2 C ε0 ε1

we have ( + ) < 2 2 2 1 (3.8) v v v ε0 C 1 ε ε1. B B U R R R~2 R~2 2 = + ≤ + ( + ) ≤ Since v is continuousS for t ST, the boundS (3.8) persists at t T1. This contradicts the maximality of T1 subject to the open condition (3.6); hence T1 T, and (3.5) is proved. < = = HARMONIC MAP FLOW FOR ALMOST-HOLOMORPHIC MAPS 13

Proposition 3.3. Let 0 R min R0, T . Suppose that v 0 satisfies (3.1) on BR 0,T , with √ < ≤ 2 2 ≥ ×[ ) (3.9) v 0 sup v t ε ε0. B U R R 0≤tgeodesic coordinates on B p , we may assume that v is defined on B 0, 1 . Then (3.9) R ( ) → ( ) 1 is preserved, and (3.1) becomes ( ) 3 2 ×[ ) ∂t ∆ v Av R Bv (3.13) Av2 1 v ( − ) ≤ + provided that R0 1 B. For R R0, the rescaled≤ ( metric+ ) ⋅ (and any further rescaling) is close enough to Euclidean that√ we have uniform volume bounds and a uniform Sobolev constant, and may apply Moser≤ ~ iteration. < 2 Let Pr x, t Br x t r , t , and write Pr Pr 0, 1 . Define (3.14) e r 1 r sup v ( ) = ( ) × [ − ] = Pr( )

and let e0, r0 be such that ( ) = ( − )

(3.15) e0 e r0 sup e r . 0≤r≤1

Choose x0, t0 Pr such that v x0, t0= (sup) = v. Letting( ) ρ0 1 r0 2, we have 0 Pr0 (3.16) ρ sup v 2e . ( ) ∈ ( 0) = 0 = ( − ) ~ Pρ0 (x0,t0)

First assume, for the sake of contradiction, that e≤0 1. Letting ρ0 ρ1 , e0 > we may deﬁne a function v1 on P1 by = 2 v1 x, t ρ1v ρ1x x0, ρ1 t 1 t0 . Then v again satisﬁes (3.13), and (3.16) implies 1 ( ) = + ( − ) + sup v1 x, t 2. P1 ( ) ≤ 14 CHONG SONG AND ALEX WALDRON

But then Moser’s Harnack inequality (see e.g [31], Lemma 2.2), applied to (3.13), gives

1 2 2 1 v1 0, 1 C v1 x, t dV dt (3.17) 0 B1 Cε = ( ) ≤ S S ( ) for a constant depending only on A. For ε≤suﬃciently small, this is a contradiction. Therefore e0 1. Now, from the deﬁnition (3.14) and (3.15), we have for any 0 r 1 ≤ e r e 1 sup v 0 . < < Pr 1 r 1 r 1 r ( ) = ≤ ≤ Then (3.13) becomes − − −

∂t ∆ v C A 1 v

on P3~4. We may again apply Moser’s( − Harnack) ≤ inequality,( + ) to ﬁnd

1 2 1 ~ sup v C v2 dV dt (3.18) P1~2 0 B3~4 ≤ C Sε. S √ By choosing ε suﬃciently small, we can≤ also ensure that sup v 1 A. Then the coeﬃcient on the RHS of (3.13) is 2, so the constant in (3.18) is independent√ of A. ≤ ~ The estimate (3.10) now follows from (3.18) by undoing the rescaling (3.12).

3.2. Epsilon-regularity theorems. The above estimates yield the following version of the standard ε-regularity theorem, as well as a split version in the case that N is K¨ahlerwith nonnegative holomorphic bisectional curvature.

Theorem 3.4. There exists ε0 0, depending on the geometry of N, and R0 0, depending on the geometry of Σ, as follows. 2 Let 0 R min R0, T and>0 τ T R . Suppose that u BR 0,T is> a solution of (1.1) such that √ < ≤ ≤ ≤ − ∶ × [ ) R (3.19) E u τ ,BR sup E u t ,UR~2 ε ε0. τ≤t

Moreover, the tension ﬁeld satisﬁes

T 2 Ck u dV dt (3.22) sup (k) u τ BR . ¼ 2+k B ×[τ+R2~2,T ) R R~2 ∫ ∫ ST ( )S Proof. By applying Kato’s inequalityS∇ toT (2.14),( )S ≤ we obtain a diﬀerential inequality of the form (3.1) with v du . The estimate (3.20) follows directly from Proposition 3.3. The derivative estimates (3.21) then follow by a standard bootstrapping argument applied to (2.12). The estimates= S (3.22)S on the tension ﬁeld can be obtained from the evolution equation

u tr du ∂t g ∂t ∇ ∇ du ∂u T ( ) = tr ∇ tr RN , du du g ∂t g ∂t ∇ N = ∆ ∇u tr+gR u , du du.

by (2.11). Taking an inner product= withT ( ) +u , we obtain(T ( ) ) 1 (3.23) ∂ ∆ u 2 T ( u) 2 u # u #du#du. 2 t By rescaling, we may( assume− )ST without( )S = − lossS∇T ( that)S +RT ( 1). ThenT ( ) we have du C ε 1 by (3.20). Then (3.23) gives an inequality √ = S S ≤ ≤ 1 (3.24) ∂ ∆ u 2 C u 2. 2 t The estimate (3.22) with k 0 now( follows− )ST from( )S Moser’s≤ ST ( Harnack)S inequality, and the higher derivative estimates are obtained by bootstrapping. = Theorem 3.5. Suppose that N is compact K¨ahlerwith nonnegative holomorphic bisectional 2 curvature, and let 0 τ min R0,T . If u Σ 0,T N solves (1.1) with 2 (3.25) < < [ ]E∂¯ u 0∶ × [δ )ε0→ then ( ( )) ≤ < 1 sup ∂u¯ Cδ 1 . Σ×[τ,T ) τ

A similar result holds for ∂u, with E∂ inS placeS ≤ of√E∂¯. +

Proof. It follows from (2.6) and (2.23) that E∂¯ u t is decreasing along the flow, so the ¯ √ assumption (3.9) is satisfied with v ∂u on any ball B τ p Σ. By applying Kato’s inequality to (2.32), we obtain an inequality of( the( )) form (3.1). The result then follows = S S ( ) ⊂ directly from Proposition 3.3, with R τ. √ Remark 3.6. By a straightforward rescaling= argument, the constant ε0 in Theorem 3.5 can be taken to equal the lowest nonzero ∂¯-energy of a harmonic 2-sphere in N. For instance, if 2 N S , we have ε0 4π. For general N, a calculation by the first-named author (to appear) 4π gives the gap value ε0 sup SH S , where HN is the holomorphic sectional curvature. = = N = 16 CHONG SONG AND ALEX WALDRON

4. Estimate on the curvature scale Let q 1. We now show that a uniform Lq bound on the stress-energy tensor

(4.1) sup S u t Lq σ ≥ τ≤t

Lemma 4.1. Given p Σ and 0 R R0, write BR BR p . Let u solve (1.1) on BR 0,T , and assume the stress-energy bound (4.1). Then for τ t1 t2 T, we have ∈ < < = ( ) ×[ ) t2 t1 E u t2 ,BR~2 E u t1 ,BR ≤ Cσ≤ <2 . R q − Proof. Choose a cutoﬀ ϕ for (BR)~2 BR≤. Integrating( ( ) by) + parts against the pointwise energy identity (2.9), we have

⊂ t2 i j e t2 ϕ dV e t1 ϕ dV ϕSij dV dt (4.2) t1 (2) C t2 t1 ϕ q−1 sup S t Lq , S ( ) − S ( ) = S S ∇ ∇ q L t1≤t≤t2 where we have applied Hölder’sinequality.≤ We( have− )Y∇ Y Y ( )Y q−1 q q q−1 2q q−1 2 (2) C 2− − ϕ q−1 dV CR q−1 q CR q . q 2 L BR R Then (4.2) yieldsY∇ Y ≤ S ≤ = −2 q (4.3) E u t2 ,BR~2 E u t1 ,BR C t2 t1 R σ as desired. ( ) − ( ( ) ) ≤ ( − ) 2 Definition 4.2. Let 0 ρ R0, 0 ε ε0, and p Σ. Given an L1 map u Bρ p N, define the (outer) energy scale λε,ρ,p u to be the minimal number λ 0, ρ such that < < < < ∈ ∶ ( ) → r (4.4) sup( )E u, Ur~2 p ε. ∈ [ ] λ then it is clear( ( )) from= the definition− ≤ that< λε,ρ,p u t 0 for ρ sufficienctly small. Conversely, suppose that λε,ρ,p u t 0 for some ρ 0 and all T ρ2 tS T.S Then, since du T ρ2 is square-integrable, we may choose R 0 such( ( )) that= ( ( )) = > E u T ρ2 ,B ε. − ≤ < ( − ) R > But, by assumption, we also have ( ( − ) ) ≤ R sup E u t ,UR~2 ε. T −ρ2≤t

↗ HARMONIC MAP FLOW FOR ALMOST-HOLOMORPHIC MAPS 17

Theorem 4.4. Let u be as above, and assume that the stress-energy bound (4.1) is satisﬁed

for some q 1. Suppose that 0 R ρ R0 and τ t1 t2 T are such that ε (4.5)≥ sup E< u ≤t ,B<ρ p E ≤u t2≤,BR< p . t1≤t≤t2 2

Then for t1 t t2, we have ( ( ) ( )) ≤ ( ( ) ( )) + q 2 ≤ ≤ Cσ t2 t (4.6) λε,ρ,p u t max 4R, . ⎡ ε ⎤ ⎢ ⎥ ⎢ ( − ) ⎥ ( ( )) ≤ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

Proof. Suppose the contrary. Then there exists t1 t t2 such that λ λε,ρ,p u t 4R satisﬁes ≤ ≤ = ( ( )) ≥ 2 2C t2 t σ (4.7) λ q . ε ( − ) Then we must have > λ (4.8) E u t ,Uλ~2 ε.

Now (4.5) and (4.8) imply ( ) ≥ λ E u t ,Bλ~2 E u t ,Bρ E u t ,Uλ~2 ε E u t ,B ε ( ( ) ) ≤ ( ( )2 R) − 2 ( ) ε ≤ E (u(t2),BR) + .− 2 But then Lemma 4.1 and (4.7) give ≤ ( ( ) ) − t2 t E u t2 ,BR E u t2 ,Bλ~4 E u t ,Bλ~2 Cσ 2 λ q ε ε − ( ( ) ) ≤ ( ) ≤ E( u( t) ,B ) + E u t ,B , 2 R 2 2 2 R < ( ( ) ) − + = ( ( ) ) 18 CHONG SONG AND ALEX WALDRON

which is a contradiction. This establishes the desired estimate (4.6). Corollary 4.5. Suppose that the stress-energy bound (4.1) holds for some q 1, and assume T . For any p Σ, there exists ρ 0 such that q ≥ (4.9) λ u t O T t 2 t T . < ∞ ∈ ε,ρ,p > Proof. Deﬁne ( ( )) = ( − ) ( ↗ ) E0 lim lim sup E u t ,Br p . r↘0 t↗T Here, the outer limit must exist because= the inner( lim( ) sup( is)) nonnegative and decreasing with respect to r. In particular, we may choose 0 ρ R0 and t1 T such that ε (4.10) sup E u t ,B< ρ

But then for any R 0, we may also choose( ( ) t1 ( t))2 ≤T such+ that ε (4.11) E u t ,B p E . > 2 R ≤ < 0 4 Combining (4.10) and (4.11), we obtain( ( ) ( )) ≥ − ε sup E u t ,Bρ p E u t2 ,BR p , t1≤t Corollary 4.6. Suppose that N is compact K¨ahlerwith nonnegative holomorphic bisectional curvature, and assume that E∂¯ u 0 ε0 or E∂ u 0 ε0. Then for each τ 0, the stress-energy tensor S t is uniformly bounded in L2 for τ t T. In particular, for each

0 ε ε0, p Σ, and T , there( exists( )) <ρ 0 such that( ( )) < > ( ) ≤ < (4.12) λ u t O T t t T . < < ∈ < ∞ ε,ρ,p > Proof. From Lemma 2.3 and Theorem( ( )) 3.5,= we( have− )( ↗ ) 1 S t 4 e t e ¯ t Cδ e t 1 ∂ ∂ ¾ ∂ τ » for τ t T. This gives S ( )S ≤ ( ) ( ) ≤ ( ) + 1 S t 2 Cδ2E u t 1 ≤ < L2(Σ) ∂ τ 1 Y ( )Y ≤ Cδ2 2δ(2 ( κ)) + 1 τ

since E∂ u t is decreasing, where we have≤ applied + (2.23). + The result now follows from the previous corollary, with q 2. ( ( )) Remark 4.7. Angenent, Hulshof, and Matano [1] ﬁrst established the bound (4.12) in the = degree-1 rotationally symmetric case originally studied by Chang, Ding, and Ye [4]. Raphael and Schweyer [18, 19] have shown that blowup occurs in that setting with the precise rate T t ` (4.13) λ t κ log T t 2`~(2`−1) ( − ) ( ) ∼ S ( − )S HARMONIC MAP FLOW FOR ALMOST-HOLOMORPHIC MAPS 19 for each ` N+, as predicted by Van den Berg, Hulshof, and King [3]. D´avila,del Pino, and Wei [5] have recently constructed more extensive examples with the ` 1 rate. ∈ = 5. Decay estimate in annular regions In this section, we obtain decay estimates for a solution of (1.1) that will apply in our setting. The results are less precise than those of [32], §5, but we give a much simpler proof.

5.1. Evolution of the angular energy. Given 0 R 1, let

1 2 (5.1) U UR r, θ R r< 1< R denote an annulus in the plane.= Let g=be{( a metric)S ≤ on≤U}of⊂ the form (5.2) g ζ2 dr2 r2dθ2 , where ζ ζ r, θ 0 is a positive smooth= function( + with) 2 2 (5.3) = ( ) > ζ r, θ 1 αr for a constant α 0. S ( ) − S ≤ For convenience, we shall work below in cylindrical coordinates s ln r, θ . Letting > 2 2 g0 ds dθ ( = ) 2 2s denote the flat cylindrical metric, we have=g ζ+e g0. The differential of u is given by (5.4) du usds= uθdθ. The tension field with respect to g0 is given= by+

0 u sus θuθ,

∗ where denotes the pullback connectionT ( ) = on∇u TN,+ ∇ as above. The heat-ﬂow equation (1.1) with respect to the metric g becomes ∇ −2 −2s (5.5) ∂tu u ζ e 0 u .

Lemma 5.1. Let u be a solution of= (1.1)T ( ) on= U 0,TT ( with) respect to a metric g given by 2 (5.2). Suppose that for some 0 η ε0, we have × [ ) 2 3 2 (5.6)

2 (5.7) f f u; r, t uθ r, θ, t dθ ¾ {r}×S1 satisﬁes a diﬀerential equality= ( ) ∶= S S ( )S 1 1 Cη (5.8) ∂ f ∂2 ∂ f Cα α 1 η. t r r r r2 − Here, ε0 depends on the geometry− + of N, −and α is the≤ constant( + of) (5.3). 20 CHONG SONG AND ALEX WALDRON

Proof. We start from the identity

1 2 2 2 2 ∂s f suθ suθ, uθ . 2 S1 By interchanging derivatives and integrating by parts, we compute = S S∇ S + a∇ f 1 2 2 2 ∂s f suθ θ sus R us, uθ us, uθ 2 S1 2 = S S∇suθS + ⟨∇su∇s, θ+uθ ( R u)s, uθ u⟩s, uθ . S1 Using the ﬂow equation (5.5),= S weS∇ getS − ⟨∇ ∇ ⟩ + ⟨ ( ) ⟩ 1 2 2 2 2 2 2s ∂s f suθ θuθ ζ e ∂tu, θuθ R us, uθ us, uθ 2 S1 On the other hand, we have = S S∇ S + S∇ S − ⟨ ∇ ⟩ + ⟨ ( ) ⟩ 1 2 ∂tf tuθ, uθ dθ ∂tu, θuθ . 2 S1 S1 Combining the above two identities, we get = S ⟨∇ ⟩ = − S ⟨ ∇ ⟩ 1 2s 2 2 2 2 (5.9) e ∂t ∂s f suθ θuθ R us, uθ us, uθ Q u , 2 S1 where ( − ) = − S S∇ S + S∇ S + ⟨ ( ) ⟩ + ( ) 2 2s Q u ζ 1 e ∂tu, θuθ . S1 Now, using (5.2) and plugging in the equation (5.5) again, we have ( ) = S ( − ) ⟨ ∇ ⟩ 2 2 −2 2s 2s Q u sup ζ ζ 1 ζ e ∂tu, θuθ α α 1 e 0 u , θuθ . S1 S1 S1 IntegratingS ( )S by≤ partsT ( and− using)T ⋅ VS thea assumption∇ (5.6),fV ≤ we( get+ ) VS ⟨T ( ) ∇ ⟩V

2s 2s Q u α α 1 e θ 0 u , uθ Cα α 1 ηe f S1 by Hölder’s inequality.S ( )S The≤ ( bound+ ) (5.6)VS also⟨∇ givesT ( )us ⟩V η≤. Hence( + ) 2 2 2 R us, uθ us, uθ CS NSη≤ f ηf S1 for η sufficiently small. Overall,VS ⟨ (5.9)( now) reads⟩V ≤ ≤ 1 2s 2 2 2 2 2 2s (5.10) e ∂t ∂s f suθ θuθ ηf Cα α 1 ηe f. 2 S1 1 Since uθ η( is small,− ) we may≤ − S alsoS∇ assumeS + S that∇ theS + image+ of the( curve+ ) u s, , t S N lies in a coordinate chart of N where the Christoffel symbol Γ is bounded by CN . Then in local coordinates,S S ≤ we have ( ⋅ ) ∶ → 2 ∂θuθ θuθ Γ uθ, uθ θuθ Γ uθ, uθ η CN η 2η. Thus S S = S∇ 2− ( )S ≤ S∇ S2+ S ( )S ≤ + ≤ θuθ ∂θuθ Γ uθ, uθ ∂ u 2 2 ∂ u Γ u , u Γ u , u 2 S∇ S = S θ θ + ( θ θ )S θ θ θ θ ∂ u 2 Cη u 2. ≥ S θ θS − S θ SS ( )S + S ( )S ≥ S S − S S HARMONIC MAP FLOW FOR ALMOST-HOLOMORPHIC MAPS 21

Then the classical Poincar´einequality on S1 yields

2 2 2 2 (5.11) θuθ uθ Cη uθ 1 Cη f . S1 S2 S1 Combining (5.10) andS (5.11),S∇ weS ≥ arriveS S atS − S S S = ( − ) 1 2s 2 2 2 2 2s (5.12) e ∂t ∂s f suθ 1 C 1 η f Cα α 1 ηe f. 2 S1 Next, note that ( − ) ≤ − S S∇ S − ( − ( + ) ) + ( + ) 1~2 1 2 2 ∂sf f∂sf suθ, uθ suθ f, 2 S1 S1 2 2 so we have ∂sf suθ =. Inserting= S into⟨∇ (5.12),⟩ ≤ weS getS∇ S 1 2s 2 2 2 2 2s (5.13) S S ≤ eS∇∂t S∂ f ∂sf 1 Cη f Cηα α 1 e f. 2∫ s On the other hand,( − ) ≤ −S S − ( − ) + ( + ) 1 ∂2 f 2 f ∂2f ∂ f 2. 2 s s s Inserting this into (5.13), we obtain ( ) = ⋅ + S S 2s 2 2s e ∂tf ∂s f 1 Cη f Cηα α 1 e . Finally, translating the above equation back to polar coordinates, we get (5.8) and the lemma − + ( − ) ≤ ( + ) is proved.

1 5.2. Construction of supersolution. Given 2 γ 1 and T 1, deﬁne the trapezoidal spacetime region ≤ < 2γ > (5.14) Ωγ r, t R, 1 0,T 1 t r 1 , as shown in Figure 1 (where γ 1 ). We will now construct a supersolution for the equation = ( 2 ) ∈ [ ] × [ )S − ≤ ≤ (5.8) on Ωγ, with controlled boundary values. Letting = 1 R,1 2γ (5.15) λγ t max R, 1 t + ,

the parabolic boundary of Ωγ is given( ) by∶= ( − ) R,1 ∂Ωγ 1 0,T λγ t , t t 0,T . 1 Let 2 γ ν 1 and consider= { the} × operator[ ) ∪ {( ( ) )S ∈ [ )} 1 ν2 (5.16)≤ < ≤ ∆ ∂2 ∂ . ν r r r r2 Let = + − ν 1 t r2ν 2γ v r, t + . 0 rν Choose µ with (( − ) + ) ( ) = ν2 (5.17) 0 µ min ν, 3ν 2 . γ < < − + 22 CHONG SONG AND ALEX WALDRON

Now let R ν ν 1 (5.18) v r, t v r, t rµ. 0 r ν2 µ2 + Lemma 5.2. On Ω , the function( ) =v r,( t satisﬁes) + + γ − (5.19) ∂ ∆ v rµ−2 ( t) ν with ( − ) ≥

(5.20) 1 sup v Cµ,ν ∂Ωγ and ≤ ≤ R,1 ν λγ t minµ,ν ν −1 (5.21) v r, t C r γ . µ,ν r ( ) Proof. We ﬁrst claim that( ) ≤ +

(5.22) ∆νv0 0. We calculate ≤ ν v 2νr2ν−1 v ∂ v 0 ν 0 r 0 2γ 1 t r2ν r = ν r2ν−1 − 1 νv0( − + ) γ 1 t r2ν r and = − 2 ( − + ) ν r2ν−1 1 ∂2v ν2v r 0 0 γ 1 t r2ν r = ν 2ν 1 r2ν−−2 ν r2ν−12νr2ν−1 1 νv ( − + ) 0 γ 1 t r2ν γ 1 t r2ν 2 r2 ( − ) + r4ν−2 ν3 − ν2 r2ν−2 + ν 2ν 1 2ν2 ν 1 νv − + 2 ( − + ) 0 1 t r2ν 2 γ2 γ 1 t r2ν γ γ r2 ( − ) + = ν2 r4ν−2 − +ν r2ν−2 ν 1 − + νv ( − + ) ν 2γ − + . 0 γ2 1 t r2ν 2 γ 1 t r2ν r2 + This gives = ( − ) − + ( − + ) − + ν2 r4ν−2 ν 2γ γ2 1 t r2ν 2 ⎡ ⎤ 1 ν2 ⎢ 2ν−2 2ν−2 ⎥ ∂2 ∂ v νv ⎢ ν r (ν − r ) ⎥ r r 2 0 0 ⎢ ( − + ) ⎥ r r ⎢ γ 1 t r2ν γ 1 t r2ν ⎥ ⎢ ⎥ ⎢ ⎥ + − = ⎢ − ν 1 1 ν + ⎥ ⎢ ⎥ ⎢ −r2+ − + ⎥ ⎢ ⎥ ν3 ⎢ r+4ν−−2 − ⎥ v⎢ + ν 2γ , ⎥ γ2 ⎣0 1 t r2ν 2 ⎦ which is non-positive for t 1, since ν= 2γ ν 1 0. This( − proves) (5.22). ( − + ) ≤ − ≤ − ≤ HARMONIC MAP FLOW FOR ALMOST-HOLOMORPHIC MAPS 23

We also have ν 2γ ν ν(2γ−ν)+νγ ν(3γ−ν) − γ − γ ∂tv ∂tv0 −ν+2γ r νr . 2ν ν 2γ 1 t −r 2γ r Note that = = ≥ − ≥ − ( − + ) µ 2 2 µ−2 ∆νr µ ν r . By (5.17), we have = ν − µ 2 ν 3γ . γ We therefore obtain − ≤ ( − ) ν(3γ−ν) − µ−2 µ−2 (5.23) ∂t ∆ν v νr γ ν 1 r r

for r 1, which establishes( (5.19).− ) ≥ − + ( + ) ≥ The bounds (5.20-5.21) follow from (5.18) by inspection. ≤ Proposition 5.3. Let 1 2 γ ν 1, and µ be as in (5.17). Suppose g r, t Ωγ R satisﬁes the equation ~ ≤ < ≤ µ−2 ( ) ∶ → ∂t ∆ν g Ar with ( − ) ≤ sup g A. ∂Ωγ Then g satisﬁes a bound ≤ R,1 ν λγ t minµ,ν ν −1 (5.24) g r, t C A r γ µ,ν r ( ) for r, t Ωγ. ( ) ≤ +

Proof.( This) ∈ follows from the previous lemma by the comparison principle, applied to the functions g and Av r, t .

( ) 5.3. Decay estimates. We can now prove a decay estimate for the angular component of the energy.

2 1 1 Lemma 5.4. Let 0 η ε0 and 2 γ 1. Let u be a solution of (1.1) on UR 0,T , with 1 T 1, where UR has a metric g as in (5.2). Suppose that for all x, t with < < ≤ < × [ ) 2γ (5.25)> max R, 1 t x 1 ( ) we have [ ( − )] ≤ S S ≤ (5.26) x du x, t x 2 du x, t x 3 (2)du x, t η.

Then for S SS ( )S + S S S∇ ( )S + S S S∇ ( )S ≤ (5.27) γ ν 1 Cη » < ≤ − 24 CHONG SONG AND ALEX WALDRON

and for all r, t as in (5.25), the angular energy (5.7) satisﬁes R,1 ν λγ t ν ν −1 (5.28) ( ) f u; r, t C η α 1 2 r γ . γ,ν r ( ) Here α is the constant of( (5.3).) ≤ ( + ) +

ν Proof. This follows from Lemma 5.1 and Proposition 5.3, where we let µ ν γ 1 and A η 1 Cα α 1 . = − We now state two versions of the decay estimate that will be used in our context. Given = ( + ( + )) 1 2 γ 1, we let t 1~2γ (5.29)~ ≤ < λR,ρ t max R, ρ 1 . γ ρ2

Theorem 5.5. Let k N. There( exists) = ε0 0, depending − on the geometry of N, and R0 0, depending on k and the geometry of Σ, as follows.5 ∈ > 2ρ > For 0 R ρ min R0, T and p Σ, let u UR~2 p 0,T N be a solution of (1.1) 2 that satisfies (4.1) for some√ q 1. Suppose that τ ρ t0 T is such that for all r, t with R,ρ < ≤2 < ∈ ∶ ( ) × [ ) → λγ t t0 ρ r 2ρ, we have > + ≤ < (5.30) E u t ,U r ε ε . ( − + ) ≤ ≤ r~2 0 Then for ν as in (5.27) and all R r ρ, we have ( ( ) ) ≤ < ν ν ν −1 R r γ 1− 1 (5.31) r1+k (k)du x, t ≤C ≤ ε C σr q , 0 γ,ν,k r ρ √ ⎛ ⎞ √ S∇ ( )S ≤ + + where x r. ⎝ ⎠ Proof. SSinceS = for R sufficiently small, radial balls in conformal and geodesic coordinates are uniformly equivalent, we may conflate the two in the statement. We rescale u by ′ ′ ′ ′ 2 ′ (5.32) u x , t u ρx , t0 ρ t 1 . Then u′ solves (1.1) on an annulus U U 1 , with α arbitrarily small in (5.3). By the ( ) = R~ρ + ( − ) r′ ε-Regularity Theorem 3.4, applied on a finite cover of U ′ by balls, (5.30) implies a bound = r ~2 sup r′ du′ x′, t′ r′2 du′ x′, t′ r′3 (2)du′ x′, t′ C ε. R~ρ,1 ′ ′ λγ (t )≤Sx S≤1 √ This is equivalent to (5.6), S with( η )S +C ε.S∇We( may)S therefore+ S∇ apply( Lemma)S ≤ 5.4, to obtain a

bound √ ν = R ′ν ν −1 f u′; r′, t′ C ε r γ . ν,γ ρr′ √ Undoing the rescaling, we obtain( ) ≤ + ν ν ν −1 R r γ (5.33) f u; r, t C ε ν,γ r ρ √ ⎛ ⎞ ( ) ≤ + 5It is again sufficient to assume that Σ has bounded⎝ geometry, or that⎠ the metric on the annulus is sufficiently close to flat. HARMONIC MAP FLOW FOR ALMOST-HOLOMORPHIC MAPS 25

R,ρ 2 for λγ t t0 ρ r ρ. It remains to bound the radial energy using the stress-energy bound (4.1). By deﬁnition (2.5), the( − stress-energy+ ) ≤ ≤ tensor is conformally invariant and has the form 1 1 S u 2 u 2 dr2 r2dθ2 2 u , u drdθ. 2 r r2 θ r θ It follows that = S S − S S ( − ) + ⟨ ⟩ 1 2 du 2 u 2 u 2 u 2 2 S . r r2 θ r2 θ √ Therefore, by H¨older’sinequalityS S = andS S (4.1),+ S weS ≤ haveS S + S S

r 2 2 E u t ,Ur~2 uθ 2 S U r r2 U r r~2 √ r~2 r ( ( ) ) ≤ 2 S S + 2S 1−S 1 S f 2 r, t dr CσrS q . r~2 r This combined with (5.33) gives ≤ S ( ) +

2ν 2ν ν −1 R r γ 21− 1 E u t ,U r C ε Cσr q . r~2 ν,γ r ρ ⎛ ⎞ ( ( ) ) ≤ + + Now the desired estimate (5.31) follows⎝ again from the ε-regularity⎠ Theorem 3.4. Theorem 5.6. Let u be as in the previous theorem, and assume that N is a Hermitian manifold. If sup ∂u¯ δ, then we have

ν ν ν −1 R r γ (5.34) S rS 1≤+k (k)du t C ε Cδr. γ,ν,k r ρ √ ⎛ ⎞ ¯ S∇ i ( )S ≤ + + Proof. We have ∂u ur r uθ dz¯ . The result⎝ follows by using⎠ this identity in place of the stress-energy bound in the previous argument. S S = S + S

6. Holder¨ continuity and bubble-tree convergence 6.1. Main theorems. Our main theorems combine the results of the previous sections, as follows.

Theorem 6.1. Let K, q 1 and k N, and ﬁx ν 0 with 1 1 2q 1 (6.1)> max∈ , > ν2 1. 2 q q − There exists ε1 0, depending on K, q, ν, and ⋅ the geometry≤ < of N, as well as R0 0, depending on k and the geometry of Σ, as follows.

Let 0 ε ε1>and Bρ Σ with >

q ε 2q−2 (6.2)< < ⊂ 0 ρ min R , T, . 0 Cσ √ < < 26 CHONG SONG AND ALEX WALDRON

Let u be a solution of (1.1) on Bρ 0,T , and assume that the stress-energy bound (4.1) holds for some 0 τ T. Suppose that τ ρ2 t T is such that × [ ) (6.3)≤ < sup E u s ,B+ ρ ≤ E

ν 1− 1 λ u t r q (6.4) ( )r≤1+k ≤ (k~)du t( ) C ε ε,ρ,p . K,q,ν,k r ρ √ ( ( )) 1 S∇1 ( )S ≤ 1 2 + Proof. Let γ max 2 , q . Then (6.1) reads γ 2 q ν , which implies 1 1 ν (6.5)= 1 1 1 ν− ν ≤ 1 . q q γ

Let n K 1, to be chosen below.− < We− may+ assume− ≤ that − 3nε1 ε0 and ν 1 3Cnε1, per (5.27). √ Now,≥ let+ 0 R ρ 2 be the minimal number such that < < −

(6.6)≤ < ~ sup E u s ,Bρ E u t ,BR 3nε. t−ρ2≤s≤t By Theorem 4.4, we have ( ( ) ) ≤ ( ( ) ) +

q Cσ t s 2 (6.7) λ3nε,ρ,p u s max 4R, ⎡ ε ⎤ ⎢ ⎥ ⎢ ( − ) ⎥ 2 ( ( )) ≤ ⎢ ⎥ for t ρ s t. But, by (6.2), we have ⎢ ⎥ ⎣ ⎦ q Cσ 2 (6.8)− ≤ ≤ ρ1−q, ε and so (6.7) implies ≤

q 1−q 2 λ3nε,ρ,p u s max 4R, ρ t s

q t s 2 (6.9) ( ( )) ≤ max 4R, ρ ( − ) ρ2 4R,ρ 2− = λγ s t ρ .

Therefore (5.30) is satisﬁed, and we conclude= from( − Theorem+ ) 5.5 that ν ν ν −1 4R r γ 1− 1 r du t C nε C σr q r ρ (6.10) √ ⎛ ⎞ √ S ( )S ≤ ν + 1− 1 + R r q 1− 1 C nε ⎝ C⎠ σr q r ρ √ √ for 4R r ρ 2, where we have≤ applied (6.5). + But, again + by (6.2), we have

− q−1 ≤ ≤ ~ σ ερ q √ √ ≤ HARMONIC MAP FLOW FOR ALMOST-HOLOMORPHIC MAPS 27

and 1− 1 1− 1 r q (6.11) σr q ε . ρ √ √ Hence (6.10) simpliﬁes to ≤

ν 1− 1 R r q (6.12) r du t C nε . r ρ √ Finally, we claim that R λS ε,ρ,p( )Su≤t 2 for the appropriate+ choice of n. Assuming the contrary, we must have R 0, λ λε,ρ,p u t ρ 2, and equality in (6.6): ≤ ( ( ))~ >sup =E u s (,B(ρ)) ≤E ~u t ,BR 3nε. t−ρ2≤s≤t Then from (6.3), we have ( ( ) ) = ( ( ) ) +

E u t ,BR 3nε E u t ,Bρ~2 Kε

and ( ( ) ) + ≤ ( ) + ρ~2 (6.13) 3n K ε E u t ,UR .

But, since λ 2R, we have ( − ) ≤ ( ) (6.14) E u t ,U 2nR U ρ 2nε. < R 2−nρ By (6.12), we also have ( ( ) ∪ ) < 1 2 2−nρ ν 1− 2−nρ R r q dr E u t ,U2nR Cnε 2nR r ρ r

−1− 1 n ( ) ≤ Cnε2S q . +

Choosing n suﬃciently large (depending≤ on q and C), we obtain 2−nρ (6.15) E u t ,U2nR ε.

Combining (6.13), (6.14), and (6.15), we ( have) ≤ ρ~2 3n K ε E u t ,UR

2nR ρ 2−nρ ( − ) ≤ E u (t ),UR U2−nρ E u t ,U2nR 2nε ε 2n 1 ε. = ( ( ) ∪ ) + ( ) Since n K 1, this is a contradiction. < + = ( + ) We have established that R λε,ρ,p u t 2, and the desired estimate (6.4) with k 0 now ≥ + follows from (6.12). The estimates for k 0 are proven similarly. ≤ ( ( ))~ = Corollary 6.2. Given 0 ε ε1, let u be> a solution of (1.1) on Σ 0,T that satisﬁes (4.1), as well as < < ε2 ×[ ) (6.16) E u τ E u t c E u τ ( ( )) − ( ( )) ≤ ( ( )) 28 CHONG SONG AND ALEX WALDRON

for τ t T. For each p Σ, k N, ρ as in (6.2), and τ ρ2 t T, we have 1− 1 λ t r q (6.17)≤ < r1∈+k (k)∈du x, t C ε ε,ρ,p+ ≤ < q,ν,k r ρ √ ( ) on Bρ p , for 2λε,ρ,p t xS∇ ρ 2.( )S ≤ +

( ) ( ) ≤ S S ≤ 1~ 1 2q−1 1 Proof. We may let ν max 2 , q q 1 q . It suﬃces to show that (6.16) implies (6.3), ½ by the following standard argument. = ⋅ > − We may assume without loss of generality that λε,ρ,p u t ρ 4, which implies

ρ (6.18) E u t ,Uρ~4 2ε.( ( )) ≤ ~ Notice that by the global energy identity ( (2.6),) the < assumption (6.16) implies T ε2 (6.19) u t 2 dV dt c . τ Σ E u τ Let ϕ be a cutoff for B B . IntegratingST ( ( ))S once against≤ ϕ in (2.9), and inserting (2.7), we ρ~2 Sρ S ( ( )) obtain 1 d ⊂ ϕ u 2 dV ϕ u 2 dV u , ϕ u . 2 dt Given t ρ2 s t, integrating from s to t and using Hölder’sinequality, we obtain S S∇ S + S ST ( )S = S ⟨T ( ) ∇ ⋅ ∇ ⟩ ε2 C ε − ≤ ϕ≤ u t 2 u s 2 dV c c ρ2E u τ E u τ ρ E u τ (6.20) √ » VS S∇ ( )S − S∇ ( )S V ≤ ε2 + » ( ( )) c ( ( )) C cε. ( ( )) E u τ √ We may assume that E u τ ε without≤ loss of generality,+ and c is sufficiently small. ( ( )) Inserting (6.18), (6.20) becomes ( ( )) ≥ (6.21) u s 2 dV ϕ u s 2 dV ϕ u t 2 dV ε u t 2 dV 3ε Bρ~2 Bρ~4 This givesS (6.3),S∇ with( )S K ≤3S andS∇ρ 2( in)S place≤ ofS ρ, whichS∇ ( )S is sufficient.+ ≤ S S∇ ( )S + 1 1− q Corollary 6.3. If u satisfies= (4.1)~ for T , then the body map u T limt↗T u t is C .

Proof. By advancing τ if necessary, we can< ∞ assume that (6.16) is( satisﬁed.) = The( result) then follows from Corollaries 4.5 and 6.2. Theorem 6.4 (Cf. Theorem 1.1). Assume that N has nonnegative holomorphic bisectional curvature, and u Σ 0,T ,T , is a smooth solution of (1.1) with E∂¯ u 0 ε0. Then u T is Cα for each α 1. The same holds∶ for× [ Struwe’s) weak< ∞ solution at each singular time T .( ( )) < ( ) < Proof. This follows by redoing the previous arguments with Theorem 5.6< ∞ in place of Theorem 5.5, and q 2 (see Corollary 4.6). 1,2 Since E∂¯ is lower-semicontinuous in the weak topology on W , we have E∂¯ u T ε0. = After restarting the ﬂow from u T , the same conclusions apply. ( ( )) < ( ) HARMONIC MAP FLOW FOR ALMOST-HOLOMORPHIC MAPS 29

Remark 6.5. Using a standard rescaling argument, one can easily replace the hypothesis that u 0 is almost holomorphic by the assumption that only holomorphic bubbles (or anti- holomorphic bubbles) appear at each singular point. However, as far as we know, this can only be( ) guaranteed by starting from an almost-holomorphic (or almost-anti-holomorphic) initial map.

6.2. Bubble-tree convergence. We can now state the bubble-tree results that follow from our main theorems. The arguments are by now standard (see Ding-Tian [6], Parker [13], or Qing-Tian [17]), once one establishes the following key estimates on the energy and oscillation in the neck region (cf. Lin-Wang [9], Lemma 3.1).

Lemma 6.6. Let 0 ρi R0, i 1,..., , be a sequence of positive numbers, and pi Σ a sequence of points. Let ui Σ 0,T N be a sequence of solutions of harmonic map ﬂow with uniformly bounded< < energy.= Suppose∞ that there exists 0 τ T such that a uniform∈ stress-energy bound of the form∶ (4.1)× [ holds,) → with q 1, as well as < < (6.22) E ui τ E ui t> δ0

for τ t T, where δ0 0 is a suﬃciently( ( )) small− ( constant( )) < (depending only on Σ,N, and the

uniform bound on the energy). Given any sequence of times ti T, let λi λε1,ρi,pi u ti , and assume≤ < that λi ρi > 0 as i . Then we have ↗ = ( ( )) βρi (6.23) lim lim lim E ui ti ,Uαλ pi 0 ~ → β↘→0 α→∞∞ i→∞ i and ( ( ) ( )) =

(6.24) lim lim lim osc βρi ui ti 0. U (pi) β↘0 α→∞ i→∞ αλi Proof. By Corollary 6.2, we have ( ) =

1− 1 λi r q (6.25) r dui x, ti C r ρi

for 2λi x ρi 2. Then (6.23-6.24)S ( follow)S by≤ integrating + (6.25).

Theorem≤ S S 6.7.≤ ~Let ui Σ 0,T N be a sequence of solutions of harmonic map ﬂow with uniformly bounded energy, and suppose that there exists 0 τ T such that (4.1), with

q 1, and (6.22) hold. Given∶ × [ any sequence) → of times ti T, there exists a subsequence (again < < labeled i) such that ui ti converges to a Hölder-continuous weak limit u∞ Σ N, smoothly away> from finitely many singular points in Σ. For each↗ singular point p, there exist finitely 2 many harmonic maps(φk) S N such that the following energy identity∶ holds:→

(6.26) lim lim E ui ti ,Br p E φk . ∶ r →0 i ↘ →∞ k 2 ( ( ) ( )) = ( ) Moreover, kφk S is connected and contains limx→p uQ∞ x .

Proof. (Sketch∪ )( It is) most natural to use the “outside-in”( argument) of Parker [13]. 30 CHONG SONG AND ALEX WALDRON

In view of the parabolic estimates of Theorem 3.4, after passing to a subsequence, we may

assume that ui ti converges smoothly away from a ﬁnite singular set, and the limit

(6.27) E0 lim lim E ui ti ,Br p 0 ( ) r↘0 i→∞ exists at each singular point p. =By the argument( ( ) of( Lemma)) ≥ 4.3, after again passing to a ○ subsequence, we may choose ρ 0 such that λi λε1,ρ,p ui ti 0. In geodesic coordinates where p 0, we may then let q be the “center-of-mass” of e u t over the ball B ○ 0 , i i i λi and put > = ( ( )) → = ( ( )) ( ) λi λε1,ρ~2,qi ui ti . ○ It follows from Corollary 6.2 and (6.27) that λi 0 for each i, and λi Cλi . Consider the rescaled sequence = ( ( )) > ≤ (6.28) vi x ui qi λix, ti . Then v x is defined for x B 0 2, and satisfies i ρ~2λi ( ) R= ( + ) (6.29) λ v 1. ( ) ∈ ( ) ⊂ε1,ρ~2λi,0 i The sequence v approaches a weak limit φ 2 p′ , , p′ N, where p′ B¯ by (6.29). i 1 R( ) = 1 n i 1 The energy inequality combined with the parabolic estimates of Theorem 3.4 imply that φ1 2∶ ∖ { ⋯ } → ∈ is a harmonic map, so in fact either E φ1, R ε0 or φ1 is constant. We claim that if φ1 is constant, then there must be at least two distinct points in the ( ) ≥ ′ singular set. Assume for contradiction that there is only one such point p1. Then since φ1 is ′ flat, all of the energy must concentrate at p1 as i ; since the origin is the center-of-mass, ′ 1 we must have p1 0. But then we clearly have E vi,U1~4 0 ε1 for i sufficiently large, → ∞ which implies that λε1,ρ~2λi,0 vi 1. This contradicts (6.29). = ( ( )) < We are left with two scenarios: either φ1 is nonconstant, with energy at least ε0, or there are at least two distinct points( ) in< the bubbling set of vi , each consuming energy at least ε0. By Lemma 6.6, we have { } ′ (6.30) E0 E φ1 lim lim E vi,Br pj r 0 i j ↘ →∞ and = ( ) + Q ( ( ))

(6.31) lim u ti, q lim φ1 x . q→p x→∞ ′ In either case, for each pj, we must have( ) = ( ) ′ (6.32) lim lim E vi,Br pj E0 ε0. r↘0 i→∞ ′ We can now rescale around each point p(j in the( same)) < fashion,− and repeat the procedure. In view of (6.32), the amount of concentrated energy at each point goes down by at least ε0, therefore the process must terminate in ﬁnitely many steps. The identity (6.30) yields the energy identity (6.26), and (6.31) yields the no-neck property. Corollary 6.8. Let u Σ 0,T N be a solution of harmonic map ﬂow, with T . Either assume that a uniform stress-energy bound (4.1) holds, or that N has nonnegative ∶ × [ ) → ≤ ∞ holomorphic bisectional curvature and E∂¯ u 0 ε0. Given any sequence of times ti T, the maps u ti converge along a subsequence in the bubble-tree sense with no necks, where u T ( ( )) < ↗ ( ) ( ) HARMONIC MAP FLOW FOR ALMOST-HOLOMORPHIC MAPS 31

is H¨oldercontinuous and the bubble maps are harmonic (or holomorphic, if E∂¯ u 0 ε0). If T , then the global weak limit u∞ of u ti is also harmonic. ( ( )) < Proof.= ∞This follows from the previous theorem.( ) For, if T , then we may choose τ as required since E u t is a nonnegative decreasing function, and let u ui for all i. The fact that the bubbles must be holomorphic in the second case follows< ∞ from the supremum bound on ∂u¯ of Theorem( ( 3.5.)) =

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Xiamen University, Fujian, China Email address: [email protected]

University of Wisconsin, Madison Email address: [email protected]