Rigidity of the harmonic map heat flow from the sphere to compact Kähler manifolds

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1 Ark. Mat., 48 (2010), DOI: /s c 2009 by Institut Mittag-Leffler. All rights reserved Rigidity of the harmonic map heat flo from the sphere to compact Kähler manifolds Qingyue Liu and Yunyan Yang Abstract. A Lojasieicz-type estimate is a poerful tool in studying the rigidity properties of the harmonic map heat flo. Topping proved such an estimate using the Riesz potential method, and established various uniformity properties of the harmonic map heat flo from S 2 to S 2 (J. Differential Geom. 45 (1997), ). In this note, using an inequality due to Sobolev, e ill derive the same estimate for maps from S 2 toacompactkähler manifold N ith nonnegative holomorphic bisectional curvature, and use it to establish the uniformity properties of the harmonic map heat flo from S 2 to N, hich generalizes Topping s result. 1. Introduction In the harmonic map heat flo, i.e. the negative gradient flo of the energy functional, the map φ beteen to Riemannian manifolds evolves according to (1.1) t φ = T (φ), φ(, 0) = φ 0, here T (φ) is the tension field of φ (cf. [2], [10]). In [11] Topping established various uniformity properties of the harmonic map heat flo beteen 2-spheres. It is interesting to see if such kind of properties hold for other manifolds. In this note e sho that his result can be generalized for flo from S 2 to compact Kähler manifolds ith nonnegative holomorphic bisectional curvature. By Mok [5], manifolds ith positive holomorphic bisectional curvature are biholomorphic to the complex projective space or an irreducible Hermitian symmetric space of rank 2, provided they are simply connected and their second Betti number is one. CP n, in particular S 2, and the torus are basic examples of manifolds ith nonnegative holomorphic bisectional curvature. This ork as partly supported by NSFC

2 122 Qingyue Liu and Yunyan Yang Recall that the key step in [11] is to get a Lojasieicz-type estimate ensuring the exponential decay of the -energy. To derive the estimate he identified S 2 ith the complex plane C via the stereographic projection, so that the Riesz potential comes into play. This trick does not ork generally, but e find that the same estimate still holds hen the targets are Kähler manifolds ith nonnegative holomorphic bisectional curvature. An interesting point of this note is that e use an inequality due to Sobolev to derive the Lojasieicz-type estimate (see Section 3 belo), hich implies the uniformity properties of the harmonic map heat flo according to Topping [11] (see Section 4 belo). So far e do not kno if such an estimate still holds for maps beteen other Kähler manifolds. The folloing question may be instructive: does any harmonic map beteen Kähler manifolds ith sufficiently small -energy have to be holomorphic? If one believes that the energies of harmonic maps beteen analytic manifolds are isolated, this must be true. Topping derived another kind of uniformity properties of the harmonic map heat flo in [12]. Ho to find its analogy beteen other manifolds is another challenge. As in [4], ho to find holomorphic spheres in Kähler manifolds via the harmonic map heat flo remains an interesting problem. Note that the key step in [9] to prove the Frankel conjecture is to find a holomorphic sphere in a suitable homotopy class. Unfortunately our convergence result need to preassume the existence of a holomorphic sphere (see Corollary 4.4 belo). A successful approach ithout preassuming the existence of a holomorphic sphere may lead to another proof of the Frankel conjecture. Acknoledgement. The authors thank the referee for valuable suggestions. 2. Notation and preliminaries In this section, e clarify some notation used in this note. Let (,g) be a closed Riemannian surface, and (N,h,J) be a compact Kähler manifold, here h is the Kähler metric, and J is the natural complex structure. Denote the Gaussian curvature of by K, and the Riemannian curvature tensor of N by R. Let φ: N be a smooth map. We rite dφ for the real differential of φ and φ: T 1,0 T 1,0 N and φ: T 1,0 T 0,1 N for the corresponding components of the complexification of dφ.

3 Rigidity of the harmonic map heat flo from the sphere to compact Kähler manifolds 123 Take holomorphic coordinate systems z=x+iy near some point p, and 1, 2,..., n near φ(p) N. In these coordinates, g and h are represented by g =2λ(z) dz d z and h =2h α β d α d β, here λ(z) is a real-valued function, h α β =h βα, and the conjugate of h α β, namely h α β =hᾱβ. Here and in the sequel the summation convention is used. Locally one can rite φ and φ as (2.1) φ= z α dz α and φ= α z dz α, here z α = α / z and z α = α / z, and similarly z ᾱ = α / z and z ᾱ = α / z. Amapφ: N is called holomorphic (anti-holomorphic) if φ=0 ( φ=0). The Hermitian inner products on T 1,0 T 0,1 andt 1,0 N T 0,1 N reads respectively. We set z, = λ(z) and z Then the energy density of φ is given by and hence the energy of φ, E(φ)= α, β e (φ)= φ 2 = λ 1 (z) α z β z h α β, e (φ)= φ 2 = λ 1 (z) α z β z hᾱβ. e(φ)= 1 2 dφ 2 = e (φ)+e (φ), e(φ) dv g = E (φ)+e (φ), = h α β, here E (φ)= e (φ) dv g and E (φ)= e (φ) dv g. Let τ α (z)= 2 ( α λ(z) z z +Γ α γδz δ γ z ), α=1, 2,..., n, here Γ α βγ are the Christoffel symbols for the Levi-Civita connection on N. The associated geometric object is τ α, and the tension field of φ is represented by α T (φ)=τ α α + τ α α.

4 124 Qingyue Liu and Yunyan Yang For some T>0, let φ: (0,T] N be a smooth solution of (1.1), hich is equivalent in holomorphic coordinates to (2.2) Let Set α t = 2 ( α λ(z) z z +Γ α γδz δ γ z ), α=1, 2,..., n. R α βγ δ = δ Γα γβ and R α βγ δ = h α ε R ε βγ δ. q 1 (φ)=λ 2 R α βγ δ( γ z δ z γ z δ z) β z α z, q 2 (φ)=λ 2 R α βγ δ( γ z δ z γ z δ z) ᾱ z β z. Then e have the folloing result (see the elliptic case in [8, p. 50]). Lemma 2.1. Let φ be a solution of (2.2). Then the partial energies e (φ) and e (φ) evolve according to ( ) (2.3) t Δ e (φ)= 2 φ 2 2q 1 (φ) 2K e (φ), ( ) (2.4) t Δ e (φ)= 2 φ 2 2q 2 (φ) 2K e (φ). Proof. For simplicity, e choose holomorphic normal coordinates near p and φ(p, t) N such that, at the point p, and at φ(p, t), λ =1, λ z =0 and λ z z = K, h α β = δ αβ, h α β,γ =Γ β αγ =0, h α β,γ δ =Γ β αγ, δ = Rβ αγ δ = R α βγ δ, h α β, γδ = Γ α βγ, δ = R α βδ γ. We obtain at p, (2.5) Δe (φ)= 2 λ ( ) 1 λ h α β z α β z z z =2h α β,z z α z β z +2h α β α zz z ᾱ z +2 α zz ᾱ z z +2 α z z ᾱ zz+2 α z ᾱ zz z +2K α z ᾱ z.

5 Rigidity of the harmonic map heat flo from the sphere to compact Kähler manifolds 125 Using the chain rule, one has (2.6) h α β,z z = h α β,γ δ γ z δ z +h α β, γδ γ z δ z. Applying the harmonic map heat equation (2.2), zz z α = z zz α = 1 2 α tz ( Γ α γδz δ γ z ) (2.7) = 1 z 2 α tz R δᾱγ μ z μ z δ γ z, and hence (2.8) ᾱ zz z = α z zz = 1 2 α t z R α δμ γ μ z δ z γ z. The covariant differential on (T 1,0 ) φ (TN) is denoted by. Then e have at p, (2.9) φ 2 = α zz ᾱ z z + α z z ᾱ zz. Obviously e have at p, (2.10) t e (φ)= α tz ᾱ z + α t z α z. Combining (2.5) (2.10), e obtain (2.3) by noting that R α βγ δ =R α δγ β =R γ βα δ. Similarly e derive (2.4). Given to J-invariant planes σ and σ in T x N, recall that the holomorphic bisectional curvature H(σ, σ ) is defined by H(σ, σ )=R(X,JX,Y,JY), here X is a unit vector in σ and Y a unit vector in σ. The folloing lemma is ell knon (see for example [8, p. 142]). Lemma 2.2. If the holomorphic bisectional curvature of (N,h) is nonnegative, i.e., H(σ, σ ) 0 for all J-invariant planes σ and σ, then in local holomorphic coordinates 1, 2,..., n, R α βγ δξ α ξβ η γ η δ 0 for all (ξ α ), (η β ) C n.

6 126 Qingyue Liu and Yunyan Yang 3. The key estimate We ould like to sho ho the folloing inequality of Sobolev can be used to derive the Lojasieicz-type estimate for Kähler manifolds ith nonnegative holomorphic bisectional curvature. Lemma 3.1. (Cf. [6, p. 385].) Let f be any smooth function ith compact support in R 2. Then f 4 dx 1 f 2 dx f 2 dx, R 2 2 R 2 R 2 here dx is the volume element of R 2. Using a covering argument e immediately have the folloing result. Lemma 3.2. Let (,g) be a compact Riemannian surface. Then there is a constant C depending only on (,g) such that for any smooth function f on there holds f 4 dv g C f 2 ( (3.1) dv g f 2 + f 2) dv g, here dv g is the standard volume element on (,g). The folloing is the key estimate (the Lojasieicz-type estimate). Lemma 3.3. Suppose N isakähler manifold of complex dimension n ith nonnegative bisectional curvature. Then there exist ε 0 >0 and C>0 such that (i) if φ: S 2 N satisfies E (φ)<ε 0, e have the estimate (3.2) E (φ) C T (φ) 2 L 2 (S 2 ) ; (ii) if φ: S 2 N satisfies E (φ)<ε 0, e have the estimate (3.3) E (φ) C T (φ) 2 L 2 (S 2 ). Proof. To begin ith, e observe that 1 (3.4) T (φ) 2 dv g = φ 2 dv g +E (φ)+ q 1 (φ) dv g, 4 S 2 S 2 S 2 here e have used the notation in Section 2. Locally q 1 (φ)=λ 2 R α βγ δ( γ z δ z γ z δ z) β z α z.

7 Rigidity of the harmonic map heat flo from the sphere to compact Kähler manifolds 127 To derive (3.4) one may integrate by parts, but e prefer the folloing approach. Let us vie φ as the initial data of the harmonic map heat flo φ(,t)(seealso[4]). Then e (φ) evolves according to (2.3) (see Lemma 2.1 above). Integrating (2.3), and noting that E (v) E (v) is homotopic invariant, d dt E (φ)= d dt E (φ)= 1 d (3.5) 2 dt E(φ)= 1 T (φ) 2 dv g, 2 S 2 e immediately get (3.4). Since N has nonnegative holomorphic bisectional curvature, e have by Lemma 2.2, q 1 (φ) Ce 2 (φ) for some positive constant C depending only on (S 2,g), and thus by (3.4) there holds 1 (3.6) T (φ) 2 dv g φ 2 dv g +E (φ) C e 2 4 (φ) dv g S 2 S 2 S 2 φ 2 dvg +E (φ) C e 2 (φ) dv g. S 2 S 2 Here e have used the fact that (3.7) φ 2 dv g φ 2 dv g, S 2 S 2 here φ is the eak gradient of the function φ. In fact, setting { φ (x), φ(x) 0, ψ(x)= 0, φ(x)=0, here is the covariant derivative on S 2, one can easily check that ψ(x) isthe eak gradient of the function φ (cf. [3, p. 150, Theorem 7.4]). If φ (x) 0, then using holomorphic normal coordinates near x and φ(x) as in the beginning of the proof of Lemma 2.1, ehave φ (x) 2 φ(x) 2. Hence (3.7) holds. Notice that Lemma 3.2 also holds for all functions belonging to the Sobolev space H 1,2 (S 2 ), applying it to f = φ, ehave ( (3.8) e 2 (φ) dv g CE (φ) E (φ)+ φ ) 2 dv g. S 2 S 2 This together ith (3.6) gives the inequality (3.2) provided that E (φ) is sufficiently small, i.e. (i) holds. In the same ay e can derive (ii).

8 128 Qingyue Liu and Yunyan Yang 4. Applications of the key estimate In this section, e give several applications of Lemma 3.3. Let (N,h,J) bea compact Kähler manifold of complex dimension n ith nonnegative holomorphic bisectional curvature. One ay to study the analytic aspects of the harmonic maps from S 2 to N is to embed N into R K for a sufficiently large integer K. This embedding is denoted by N R K.Letv(,t): S 2 [0, ) N R K be the harmonic map heat flo evolving according to (1.1), hich is equivalent to v (4.1) t =Δ S2v+A(v)( v, v), here Δ S 2 is the Laplace Beltrami operator on S 2,andA(v)(X, Y ) (T v N) is the second fundamental form of N. We ill denote the right-hand side of (4.1) also by T, ithout confusion. According to Strue [10], (4.1) has a global eak solution and is smooth except for finitely many points in space-time. The folloing theorem is about the bubbling phenomenon in the harmonic map flo, and no ell knon (cf. [1], [7] and[13]). Theorem 4.1. Let v be a solution of (4.1). Then there exist finitely many points {ˆx k } m k=0, and finitely many nonconstant harmonic maps { ω k} s k=0 from S2 to N R K together ith (i) sequences {t i } i=1 ith t i ; (ii) v(,t i ) ω 0, eakly in W 1,2 (S 2, R K ) and strongly in W 1,2 loc (S2 \{ˆx 1,..., ˆx m }, R K ) as i ; (iii) s lim E(v(,t)) = E(ω k ). t The map ω 0 : S 2 N is called body map, and the maps ω i : S 2 N are called bubble maps. Once the key estimate (Lemma 3.3) is established, as pointed out by Topping, one can get the uniformity properties of the harmonic map heat flo [11, Theorems 2 and 3]. Theorem 4.2. Suppose e have a solution v of the harmonic map heat flo from S 2 [0, ) N R K, here N is a compact Kähler manifold ith nonnegative holomorphic bisectional curvature. Suppose moreover that at infinite time, the bubble maps and the body map defined in Theorem 4.1 are all holomorphic or anti-holomorphic. Then (i) v(,t) ω 0, as t, eakly in W 1,2 (S 2, R K ), and hence strongly in L p (S 2, R K ) for any p [1, ); k=0

9 Rigidity of the harmonic map heat flo from the sphere to compact Kähler manifolds 129 (ii) v(,t) ω 0 uniformly as t in Cloc k (S2 \{ˆx 1,..., ˆx m }) for any k N; (iii) for any r>0 sufficiently small and k {1,..., m}, the quantity E (ˆxk,r)(v(,t)) = e(v(,t)) dv g B r(ˆx k ) converges to a limit F k,r uniformly as t, here B r (ˆx k ) is the geodesic ball centered at ˆx k S 2 ith radius r. Theorem 4.3. Let N be a compact Kähler manifold ith nonnegative bisectional curvature. Suppose e have to solutions of the heat equation (4.1), hich e rite as maps v and from S 2 [0, ) to N R K, ith initial maps v 0 and 0 from S 2 to N R K. Suppose moreover that for the flo v there are no bubble maps at finite time and that the bubble maps and the body map at infinite time are all holomorphic or all anti-holomorphic. Then ith x k being the blo-up points of the flo v, e have that for all ε>0, Ω S 2 \{ˆx 1,..., ˆx m } and r>0 sufficiently small, there exists δ>0 independent of such that if v 0 0 W 1,2 (S 2 ) <δ, then (i) v(,t) (,t) L 2 (S 2 )<ε for all t>0; (ii) v(,t) (,t) W 1,2 (Ω)<ε for all t>0. The hypothesis that all bubbles are holomorphic (anti-holomorphic) is to guarantee that E (v(,t i )) 0(E (v(,t i )) 0), as i, hich can be easily derived from Ding Tian s description of bubbles in Theorem 4.1 (cf. [1]) instead of Theorem 1 in [11]. No Topping s argument in [11] is easily seen to be valid if e use Lemma 3.3 instead of Lemma 1 in [11]. Since no ne idea comes out, e omit the details of the proofs of Theorems 4.2 and 4.3 here but refer the reader to Section 2 in [11]. Finally e state a special case of Theorem 4.3. Corollary 4.4. Let N R K be a compact Kähler manifold ith nonnegative holomorphic bisectional curvature. Suppose u is a holomorphic map from S 2 to N. For any ε>0 there exists δ>0 such that if v 0 is another map from S 2 to N ith (4.2) v 0 u W 1,2 (S 2 ) <δ,

10 130 Qingyue Liu and Yunyan Yang: Rigidity of the harmonic map heat flo from the sphere to compact Kähler manifolds then v(,t), the solution of the harmonic map heat flo (4.1) ith initial map v 0, exists globally and converges to a holomorphic map v. Moreover, e have (4.3) v(,t) u W 1,2 (S 2 ) <ε for all t>0. Again e refer the reader to [11] for the proof of Corollary 4.4. References 1. Ding,W.andTian, G., Energy identity for approximate harmonic maps from surfaces, Comm. Anal. Geom. 3 (1995), Eells, J. andsampson, J., Harmonic maps of Riemann manifolds, Amer. J. Math. 86 (1964), Gilbarg, D.andTrudinger, N.,Elliptic Partial Differential Equations of Second Order, Springer, Berlin, Liu, Q., Rigidity of the harmonic map heat flo, Preprint, Mok, N., The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature, J. Differential Geom. 27 (1988), Pachpatte, B.,Mathematical Inequalities, North-Holland Mathematical Library 67, Elsevier, Amsterdam, Qing, J.andTian, G., Bubbling of the heat flos for harmonic maps from surfaces, Comm. Pure Appl. Math. 50 (1997), Schoen, R.andYau,S.T.,Lectures on Harmonic Maps, Conference Proceedings and Lecture Notes in Geometry and Topology II, International Press, Cambridge, Siu, Y.T.andYau,S.T.,CompactKähler manifolds of positive bisectional curvature, Invent. Math. 59 (1980), Strue, M., On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Helv. 60 (1985), Topping, P., Rigidity in the harmonic map heat flo, J. Differential Geom. 45 (1997), Topping, P., Repulsion and quantization in almost-harmonic maps, and asymptotics of the harmonic map flo, Ann. of Math. 159 (2004), Wang, C., Bubble phenomenon of certain Palais Smale sequences from surfaces to general targets, Houston J. Math. 22 (1996), Qingyue Liu School of Mathematical Sciences Peking University Beijing P. R. China qyliu@amss.ac.cn Received February 1, 2008 published online April 28, 2009 Yunyan Yang Department of Mathematics Renmin University of China Beijing P. R. China yunyanyang@ruc.edu.cn