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NONDEGENERACY OF HALF-HARMONIC MAPS FROM R INTO S1 7 1 YANNICK SIRE, JUNCHENG WEI, AND YOUQUAN ZHENG 0 2 n Abstract a J We prove that the standard half-harmonic map U : R → S1 defined 3 by 1 x2−1 P] x → x−22+x1 (cid:18)x2+1(cid:19) A is nondegenerate in the sense that all bounded solutions of the . h linearized half-harmonic map equation are linear combinations of at threefunctionscorresponding torigidmotions(dilation, translation m and rotation) of U. [ 1 v 1. Introduction 9 2 Due to their importance in geometry and physics, the analysis of 6 3 critical points of conformal invariant Lagrangians has attracted much 0 attention since 1950s. A typical example is the Dirichlet energy which . 1 is defined on two-dimensional domains and its critical points are har- 0 monic maps. This definition can be generalized to even-dimensional 7 1 domains whose critical points are called polyharmonic maps. In recent : v years, people are very interested in the analog of Dirichlet energy in Xi odd-dimensional case, for example, [2], [3], [4], [5], [13], [14] and the references therein. Among these works, a special case is the so-called r a half-harmonic maps from R into S1 which are defined as critical points of the line energy 1 L(u) = |(−∆R)14u|2dx. (1.1) 2 R Z Note that the functional L is invariant under the trace of conformal maps keeping invariant the half-space R2: the Möbius group. Half- + harmonic maps have close relations with harmonic maps with partially free boundary and minimal surfaces with free boundary, see [12] and [13]. Computing the associated Euler-Lagrange equation of (1.1), we obtain that if u : R → S1 is a half-harmonic map, then u satisfies the 1 2 Y. SIRE,J. WEI, ANDY.ZHENG following equation, 1 |u(x)−u(y)|2 (−∆R)21u(x) = dy u(x) in R. (1.2) 2π |x−y|2 R (cid:18) Z (cid:19) It was proved in [13] that Proposition 1.1. ([13]) Let u ∈ H˙ 1/2(R,S1) be a non-constant entire half-harmonic map into S1 and ue be its harmonic extension to R2. + Then there exist d ∈ N, ϑ ∈ R, {λ }d ⊂ (0,∞) and {a }d ⊂ R k k=1 k k=1 such that ue(z) or its complex conjugate equals to d λ (z −a )−i eiϑ k k . λ (z −a )+i k k k=1 Y Furthermore, 1 E(u,R) = [u]2 = |∇ue|2dz = πd. H1/2(R) 2 R2 Z + This proposition shows that the map U : R → S1 x2−1 x → x2+1 −2x (cid:18)x2+1(cid:19) is a half-harmonic map corresponding to the case ϑ = 0, d = 1, λ = 1 1 and a = 0. In this paper, we prove the nondegeneracy of U which is 1 a crucial ingredient when analyzing the singularity formation of half- harmonicmapflow. NotethatU isinvariantunder translation, dilation cosα −sinα androtation, i.e., forQ = ∈ O(2),q ∈ Randλ ∈ R+, sinα cosα (cid:18) (cid:19) the function x−q cosα −sinα x−q QU = U λ sinα cosα λ (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) still satisfies (1.2). Differentiating with α, q and λ respectively and then set α = 0, q = 0 and λ = 1, we obtain that the following three functions 2x −4x −4x2 Z (x) = x2+1 , Z (x) = (x2+1)2 , Z (x) = (x2+1)2 (1.3) 1 x2−1 2 2(1−x2) 3 2x(1−x2) (cid:18)x2+1(cid:19) (x2+1)2! (x2+1)2 ! satisfy the linearized equation at the solution U of (1.2) defined as 1 |U(x)−U(y)|2 1 (−∆R)2v(x) = dy v(x) 2π |x−y|2 R (cid:18) Z (cid:19) 1 (U(x)−U(y))·(v(x)−v(y)) + dy U(x) (i1n.4R) π |x−y|2 R (cid:18) Z (cid:19) NONDEGENERACY OF HALF-HARMONIC MAPS FROM R INTO S13 for v : R → T S1. Our main result is U Theorem 1.1. The half-harmonic map U : R → S1 x2−1 x → x2+1 −2x (cid:18)x2+1(cid:19) is nondegenerate in the sense that all bounded solutions of equation (1.4) are linear combinations of Z , Z and Z defined in (1.3). 1 2 3 In the case of harmonic maps from two-dimensional domains into S2, the non-degeneracy of bubbles was proved in Lemma 3.1 of [7]. Integro-differential equations have attracted substantial research in recent years. The nondegeneracy of ground state solutions for the fractional nonlinear Schro¨dinger equations has been proved by Frank and Lenzmann [10], Frank, Lenzmann and Silvestre [11], Fall and Valdinoci [9], and the corresponding result in the case of fractional Yamabe prob- lem was obtained by Dávila, del Pino and Sire in [6]. 2. Proof of Theorem 1.1 The rest of this paper is devoted to the proof of Theorem 1.1. For convenience, we identify S1 with the complex unite circle. Since Z , 1 Z and Z are linearly independent and belong to the space L∞(R)∩ 2 3 Ker(L ), weonlyneedtoprovethatthedimensionofL∞(R)∩Ker(L ) 0 0 is 3. Here the operator L is defined as 0 1 |U(x)−U(y)|2 1 L0(v) = (−∆R)2v(x)− 2π |x−y|2 dy v(x) R (cid:18) Z (cid:19) 1 (U(x)−U(y))·(v(x)−v(y)) − dy U(x), π |x−y|2 R (cid:18) Z (cid:19) 4 Y. SIRE,J. WEI, ANDY.ZHENG for v : R → T S1. Let us come back to equation (1.4), for v : R → U T S1, v(x)·U(x) = 0 holds pointwisely. Using this fact and the defini- U 1 tion of (−∆R)2 (see [8]), we have 1 |U(x)−U(y)|2 1 (−∆R)2v(x) = dy v(x) 2π |x−y|2 R (cid:18) Z (cid:19) 1 (U(x)−U(y))·(v(x)−v(y)) + dy U(x) π |x−y|2 R (cid:18) Z (cid:19) 1 |U(x)−U(y)|2 = dy v(x) 2π |x−y|2 R (cid:18) Z (cid:19) 1 (U(x)−U(y)) + dy ·v(x) U(x) π |x−y|2 R (cid:18) Z (cid:19) 1 (v(x)−v(y)) + dy ·U(x) U(x) π |x−y|2 R (cid:18) Z (cid:19) 1 |U(x)−U(y)|2 = dy v(x) 2π |x−y|2 R (cid:18) Z (cid:19) 1 (v(x)−v(y)) + dy ·U(x) U(x) π |x−y|2 R (cid:18) Z (cid:19) 1 |U(x)−U(y)|2 = dy v(x) 2π |x−y|2 R (cid:18) Z (cid:19) 1 + (−∆R)2v(x)·U(x) U(x). (cid:16) (cid:17) Therefore equation (1.4) becomes to 1 |U(x)−U(y)|2 1 1 (−∆R)2v(x) = dy v(x)+ (−∆R)2v(x)·U(x) U(x) 2π |x−y|2 R (cid:18) Z (cid:19) (cid:16) (cid:17) 2 1 = v(x)+ (−∆R)2v(x)·U(x) U(x). (2.1) x2 +1 (cid:16) (cid:17) Next, wewill liftequation(2.1)toS1 viathestereographic projection from R to S1 \{pole}: 2x S(x) = x2+1 . (2.2) 1−x2 (cid:18)x2+1(cid:19) It is well known that the Jacobian of the stereographic projection is 2 J(x) = . x2 +1 For a function ϕ : R → R, define ϕ˜ : S1 → R by ϕ(x) = J(x)ϕ˜(S(x)). (2.3) NONDEGENERACY OF HALF-HARMONIC MAPS FROM R INTO S15 Then we have 1 ϕ˜(S(x))−ϕ˜(S(y)) 1 [(−∆S1)2ϕ˜](S(x)) = dS(y) π |S(x)−S(y)|2 R Z 1 1+x2ϕ(x)− 1+y2ϕ(y) 2 = 2 2 dy π 4(x−y)2 1+y2 R Z (x2+1)(y2+1) 1+x2 (1+x2)ϕ(x)−(1+y2)ϕ(y) = dy 4π (x−y)2 R Z 1+x2 x2 +1 = (−∆R)1/2 ϕ(x) 2 2 (cid:20) (cid:21) 1+x2 = (−∆R)1/2[ϕ˜(S(x))]. 2 Therefore, (−∆R)1/2[ϕ˜(S(x))] = J(x)[(−∆S1)12ϕ˜](S(x)). Denote v = (v ,v ) and let v˜ , v˜ be the functions defined by (2.3) 1 2 1 2 respectively. Then the linearized equation (2.1) becomes J(x)(−∆S1)21v˜1 = J(x)v˜1 + xx22+−11xx22+−11J(x)(−∆S1)21v˜1 + xx22−+11x−22+x1J(x)(−∆S1)21v˜2, ( J(x)(−∆S1)12v˜2 = J(x)v˜2 + x−22+x1xx22+−11J(x)(−∆S1)21v˜1 + x−22+x1x−22+x1J(x)(−∆S1)21v˜2. Since J(x) > 0 and set U = (cosθ,sinθ), we get (−∆S1)12v˜1 = v˜1 +cos2θ(−∆S1)21v˜1 +cosθsinθ(−∆S1)21v˜2, (cid:26) (−∆S1)12v˜2 = v˜2 +cosθsinθ(−∆S1)21v˜1 +sin2θ(−∆S1)12v˜2, which is equivalent to 1 1 1 (−∆S1)2v˜1 = 2v˜1 +cos2θ(−∆S1)2v˜1 +sin2θ(−∆S1)2v˜2, 1 1 1 (cid:26) (−∆S1)2v˜2 = 2v˜2 +sin2θ(−∆S1)2v˜1 −cos2θ(−∆S1)2v˜2. Set w = v˜ +iv˜ , z = cosθ+isinθ, then we have 1 2 (−∆S1)21w = 2w+z2(−∆S1)21w¯. (2.4) Here w¯ is the conjugate of w. Since v ∈ L∞(R), w is also bounded, so we can expand w into fourier series ∞ w = a zk. k k=−∞ X 6 Y. SIRE,J. WEI, ANDY.ZHENG 1 Note that all the eigenvalues for (−∆S1)2 are λk = k, k = 0,1,2,·· ·, see [1]. Using (2.4), (−∆S1)12zk = kzk and (−∆S1)21z¯k = kz¯k, we obtain (−k −2)ak = (2−k)a¯2−k, if k < 0, (k −2)ak = (2−k)a¯2−k, if 0 ≤ k ≤ 2,   ak = a¯2−k, if k ≥ 3. Furthermore, fromthe orthogonal condition v(x)·U(x) = 0 (so (v˜ ,v˜ )·  1 2 (cosθ,sinθ) = 0), we have ak = −a¯2−k, k = ···−1,0,1,··· . Thus a = 0, if k < 0 or k ≥ 3 k and a = −a¯ , a = −a¯ 0 2 1 1 hold, which imply that i (z2 −1) w = −a¯ +a z +a z2 = a(iz)+b (z −1)2 +c . 2 1 2 2 2 (cid:20) (cid:21) Here a, b, c are real numbers and satisfy relations c i i(a−b) = a , + b = a . 1 2 2 2 And it is easy to check that iz, i(z − 1)2 and (z2−1) are respectively 2 2 Z , Z and Z under stereographic projection (2.2). By the one-to-one 1 2 3 correspondence of w and v, we know that the dimension of L∞(R) ∩ Ker(L ) is 3. This completes the proof. 0 References [1] Sun-Yung A. Chang and Paul C. Yang. Extremal metrics of zeta function determinants on 4-manifolds. Ann. of Math. (2), 142(1):171–212,1995. [2] Francesca Da Lio. 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[14] ArminSchikorra.Regularityofn/2-harmonicmapsintospheres.J.Differential Equations, 252(2):1862–1911,2012. JohnsHopkinsUniversity,Departmentofmathematics,KriegerHall, Baltimore, MD 21218, USA E-mail address: [email protected] Department of Mathematics, University of British Columbia, Van- couver, B.C., Canada, V6T 1Z2 E-mail address: [email protected] SchoolofScience,TianjinUniversity,92WeijinRoad,Tianjin300072, P.R. China E-mail address: [email protected]