Stable Stationary Harmonic Maps to Spheres

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1 Acta Mathematica Sinica, English Series Published online: Dec., 5 DOI: 1.17/s Stable Stationary Harmonic Maps to Spheres Fang Hua LIN Courant Institute of Mathematical Sciences, New York University, NY 11, USA linf@cims.nyu.edu Chang You WANG Department of Mathematics, University of Kentucky, Lexington, KY 4513, USA cywang@ms.uky.edu Dedicated to Professor WeiYue Ding on the occasion of his 6th birthday Abstract For k 3, we establish new estimate on Hausdorff dimensions of the singular set of stable-stationary harmonic maps to the sphere S k. We show that the singular set of stable-stationary harmonic maps from B 5 to S 3 is the union of finitely many isolated singular points and finitely many Hölder continuous curves. We also discuss the minimization problem among continuous maps from B n to S. Keywords stable stationary harmonic map, Hausdorff dimension, rectifiablity MR() Subject Classification 58E, 35J5 1 Introduction Let Ω R n be a bounded, smooth domain, S k = {x R k+1 : x =1} be a unit sphere in R k+1. The Sobolev space H 1 (Ω, S k )={v H 1 (Ω, R k+1 ):v(x) S k for a.e. x Ω}. Recall that u H 1 (Ω, S k ) is a harmonic map, if Δu + u u =, in D (Ω). (1.1) It is well known that Schoen Uhlenbeck [1, ] have proved that if u H 1 (Ω, S k )isan energy minimizing harmonic map, then the Hausdorff dimension of S (u), the singular set of u, isatmostn d(k) 1, with, k =, d(k) = 3, {[ ] } k =3, k min +1, 6, k 4, where [ ] is the integer part. This theorem has been extended to stable stationary harmonic maps u H 1 (Ω, S k ), k 3, by Hong Wang [3]. One of the purposes of this note is to show that the theorems in [] and [3] can be improved for 4 k 7. In order to describe it, we recall Definition 1 A harmonic map u H 1 (Ω, S k ) is a stationary harmonic map, if { n } u div(x) i u, j u i X j dx =, X C(Ω, 1 R n ). (1.) Ω i,j=1 Received August 31, 5, Accepted September 31, 5 Both authors are partially supported by NSF

2 Lin F. H. and Wang C. Y. Examples of stationary harmonic maps include energy minimizing harmonic maps and smooth harmonic maps. The crucial property of stationary harmonic maps is the energy monotonicity identity (cf. Price [4] or Schoen [5]) : For x Ω, <r R<d(x, Ω), R n u r n u = y x n u y x. (1.3) Definition or equivalently B R (x) B r (x) B R (x)\b r (x) A harmonic map u H 1 (Ω, S k ) is a stable harmonic map, if d ( ) dt u + tφ t= dx, (1.4) u + tφ Ω Ω { φ u φ } dx (1.5) for any φ C (Ω, R k+1 ),with φ(x),u(x) =for a.e. x Ω. It follows from the proof of [] Theorem.7 thatfork 3, (1.5) implies { η k } Ω k u η dx, η C (Ω). (1.6) Now we define 3, k =3 4, k =4 ˆd(k) = 5, 5 k 9 6, k 1. By direct comparison, we have that ˆd(k) =d(k) fork =3andk 8, ˆd(4) = 4 = d(4) + 1, ˆd(5) = 5 = d(5) +, ˆd(6) = 5 = d(6) + 1, ˆd(7) = 5 = d(7) + 1. Now we are ready to state the first theorem. Theorem 1 For k 3, letu H 1 (Ω, S k ) be a stable stationary harmonic map. Then the singular set of u, S (u), has its Hausdorff dimension at most n ˆd(k) 1. From the compactness theorem on stable stationary harmonic maps into S k (k 3) in [3] and the dimension reduction argument in [1, ], Theorem 1 follows from the nonexistence of stable stationary tangent maps in R m for m ˆd(k). To show this fact, we need to establish a sharp Kato s type inequality for smooth harmonic maps from S m to S k (see Proposition.3 below). We remark that such an inequality has been proved for harmonic functions (see Yau [6]), and for harmonic maps from S 3 to S 3 recently by Nakajima [7]. Moreover, It has been proved by [7] that any tangent map at an isolated singular point of a stable stationary harmonic map from B 4 to S 3 is, after isometry transformations, of the form x x (see also Theorem 3.1 below). On the basis of this type of classifications, we are able to generalize the theorems by Hardt Lin [8] (see also Almgren Lieb [9]) on minimizing harmonic maps from B 3 to S and prove Theorem Assume that u H 1 (B 4, S 3 ) is a stable stationary harmonic map. Then for any <ɛ<1, there exist positive constants K ɛ,l ɛ such that (i) r n u K ɛ, B r (a) B 1 ɛ, (1.7) B r (a) and (ii) the number of singularities of u in B 1 ɛ is bounded by L ɛ. We are also able to extend the structure theorem on the singular set of minimizing harmonic maps from B 4 to S by Hardt Lin [1] and obtain Theorem 3 Assume that u H 1 (B 5, S 3 ) is a stable stationary harmonic map. Then for any ɛ>, the singular set of u inside B 1 ɛ is the union of finitely many isolated singular points

3 Stable Stationary Harmonic Maps to Spheres 3 and a finite collection of bi-hölder continuous curves with a finite number of intersections. Moreover, the number of isolated singular points, curves, and the intersection points is bounded by a universal constant M ɛ >. The paper is written as follows. In, we outline the proof of Theorem 1. In 3, we prove Theorems and 3. Since the stability condition for stable harmonice maps into S is void, it remains unknown whether Theorem 1 remains to be true for k =. In 4, we consider a class of harmonic maps which are weak limits among continuous maps to S, and study their regularity properties. Kato s Inequality and the Proof of Theorem 1 In this section, we prove a sharp version of Kato s type inequality and combine it with both the stablity inequality (.1) and the Bochner formula (.) to show Theorem 1. We start with Proposition.1 For m and k 3, letφ H 1 (S m, S k ) be a harmonic map such that φ(x) =φ( x x ):Rm+1 S k is a stable harmonic map. Then we have { η (m 1) + η k } S 4 k φ η, η C (S m ). (.1) m Proof Note that φ is homogeneous of degree zero. For any η C (S m ), we let η(x) = η 1 ( x )η ( x x ), where η 1 C (R + )satisfies (η 1(r)) r m { dr η 1 (r) r m dr =inf (η (r)) r m } dr η(r) r m dr : η (m C 1) ((, + )) =, 4 and substitute it into (1.6). Then a straightforward calculation implies (.1). Now we recall the Bochner formula for smooth harmonic maps (see Eells Lemaire [11]). Proposition. For m, k, ifφ C (S m, S k ) is a harmonic map, then ( ) 1 Δ φ = φ +(m 1) φ { eα φ eβ φ eα φ, eβ φ }, (.) α,β=1 where {e α } m is any local orthonormal frame of S m. The crucial step to improve d(k) is the following Kato s type inequality for harmonic maps (see Nakajima [7] for m = k =3). Proposition.3 If φ C (S m, S k ) is a nonconstant harmonic map, then φ m m 1 φ, x S m. (.3) Moreover, the equality holds at a point x S m iff φ(x) =. Proof By choosing normal coordinates at x S m and φ(x ) S k,wehave k φ (x )= (φ i αβ(x )). i=1 α,β=1 On the other hand, since φ is a harmonic map, we have φ i αα(x )=, 1 i k. For any 1 i k, let{λ i α} 1 α m R be the eigenvalues of (φ i αβ (x )) such that λ i 1 λ i m. Thenwehave k φ (x )= (λ i α), (.4) and i=1 λ i α =, 1 i k. (.5)

4 4 Lin F. H. and Wang C. Y. On the other hand, by the Cauchy Schwarz inequality and (.5), we have m 1 (λ i α) 1 ( m 1 ) λ i α = (λi m), 1 i k, (.6) m 1 m 1 so that (λ i α) m m 1 (λi m). (.7) By the Releigh quotient, we have, for 1 i k, λ i m = sup { v R m } m ( m β=1 φi αβ (x )v β ) v. Therefore, we have φ i (x ) φ i (x ) m [ m φ i m 1 αβ(x )φ i β(x )], 1 i k. (.8) β=1 Taking sum of (.8) over i and applying the Cauchy Schwarz inequality and the Minkowski inequality, we have, at x, φ φ `= ( k )( k ) ( k φ i φ i i=1 i=1 i=1 m { k [ m ( m ) ] 1 } φ i m 1 αβφ i β m m 1 i=1 [ k i=1 β=1 β=1 φ i αβφ i β ) φ i φ i ] = m m 1 φ, φ. Since φ = φ, φ φ, this yields (1.7). Observe that the equality in (1.7) holds at x S m if and only if both the Cauchy Schwarz inequality and the Minkowski inequality are equalities. This implies: (i) λ i 1 = = λ i m 1 = λi m m 1, 1 i k, (ii) φ 1 φ 1 = = φ k φ k, (iii) φ i (x ) is an eigenfunction of φ i (x ) with the eigenvalue λ i m: φ i αβ(x )φ i β(x )=λ i mφ i α(x ), 1 i k, 1 α m, β=1 and (iv) m β=1 φi αβ (x )φ i β (x )(= λ i mφ i α(x )) is independent of 1 i k and 1 α m. In particular, we have that φ i (x )=μ i (1,...,1) for some μ i for1 i k. Notethat (i) and (ii) imply (λ 1 m) (μ 1 ) = = (λk m) (μ k ), (.9) and (iii) implies λ 1 mμ 1 = = λ k mμ k. (.1) (.9) and (.1) imply λ i 1 = = λ i m. Hence we have λ i α =forall1 α m, 1 i k, i.e. φ(x ) =. This completes the proof of Proposition.3.

5 Stable Stationary Harmonic Maps to Spheres 5 Combining Proposition. with Proposition.3, we have Proposition.4 If φ C (S m, S k ) is a nonconstant harmonic map, then ( ) 1 Δ φ m m 1 φ +(m 1) φ + 1 m m φ 4. (.11) Furthermore, the equality holds at a point x S m iff φ(x) =, eα φ (x) eβ φ == eα φ, eγ φ (.1) for all 1 α, β, γ m, α γ. Proof Observe that the lower bound of the last term in (.) is given by { e α φ e β α φ, β φ } α φ 4 φ 4 α,β=1 1 ( m ) α φ φ 4 m 1 m m φ 4. (.13) Therefore (.) and (.3) imply (.11). Moreover, the equality in (.11) holds iff both (.3) and (.13) are equalities and hence (.1) is true. This proves Proposition.4. Now we are ready to show the obstruction for the existence of stable tangent maps. Recall that a nonconstant map φ C (S m, S k ) is called a stable tangent map, if φ is a harmonic map and its homogeneous degree zero extension φ(x) =φ( x x ):Rm+1 S k is a stable harmonic map. Proposition.5 For m and k 3, ifφ C (S m, S k ) is a stable tangent map, then (m 1) 4m k k. Proof By direct calculations, we have ( ) 1 Δ φ = φ Δ φ + φ. This, combined with (.11), implies Δ φ 4 m 1 φ 1 +(m 1) ( φ 1m ) φ 3. (.14) Integrating (.14) over S m,weobtain φ 1 (m 1) (m 1) + φ φ 3. (.15) S 4 4m m S m On the other hand, the stablity inequality (.1) implies φ 1 (m 1) + φ k φ 3. (.16) S 4 k m S m Therefore we have ( ) k (m 1) φ 3. k 4m S m This implies k k (m 1) 4m. The proof is complete. It is easy to check that if m< ˆd(k) then (m 1) 4m following non-existence theorem on stable tangent maps. Proposition.6 < k k. Therefore we obtain the For k 3, there exists no stable tangent map in R m+1 for m ˆd(k) 1. Completetion of Proof of Theorem 1 By the compactness theorem on stable stationary harmonic maps into S k (k 3) in [3] and Proposition.6, Theorem 1 follows from the Federer dimension reduction argument as in [].

6 6 Lin F. H. and Wang C. Y. 3 Proof of Theorems and 3 In this section, we first review the classification theorem in [7] on stable tangent maps from R 4 to S 3, which also follows from Propositions.4 and.5 above, and then we outline proofs of Theorems and 3, which are natural extensions of the corresponding theorems on minimizing harmonic maps from B 3 to S by Hardt Lin [8] and Almgren Lieb [9], and from B 4 to S by Hardt Lin [1]. Theorem 3.1 ([7]) Suppose that φ C (S 3, S 3 ) is a nontrivial harmonic map such that its homogeneous of degree zero extension φ(x) =φ( x x ):R4 S 3 is a stable harmonic map. Then ( ) x φ(x) =Q (3.1) x for some orthogonal rotation Q O(3). Proof The reader can refer to [7] for the orginal proof. Here we indicate a slightly different argument that follows from Propositions.4 and.5. In fact, when m = k = 3wehave k k = (m 1) 4m. Therefore both (.3) and (.11) must be equalities and (.1) is satisfied for any x S 3.Inparticular,φ: S 3 S 3 is a conformal, totally geodesic map. This is possible iff φ : S 3 S 3 is an isometry. A direct consequence of Theorem.1 is the following fact on stable stationary harmonic maps from R 4 to S 3. Proposition 3. For a bounded domain Ω R 4,ifu H 1 (Ω, S 3 ) is a stable stationary harmonic map, then S (u), the singular set of u, is discrete, and for any x S, Θ(u, x ) = lim r u r B = 3 r (x ) S3. (3.) Moreover, if there exists an r > such that B r (x ) u = 3 S3 r, then there exists a Q O(3) such that u(x) =Q ( x x x x ) for x Br (x ). Proof The discreteness of S (u) follows from the partial regularity theorem in [3]. Moreover, it follows from the monotonicity identity (1.3) and the compactness theorem in [3] that for any x S (u) andr i, there exists a nontrivial smooth harmonic map φ C (S 3, S 3 )such that, after taking possible subsequences, ( lim x H i u(x + r i x) φ =, R>. (3.3) x ) 1 (B R ) In particular, φ( x x ):R4 S 3 is a stable harmonic map. Therefore, Theorem 3.1 implies φ = Q for some Q O(3). Hence ( )) Θ(u, x )= x (Q = B 1 x x 1 B 1 x =3 rdr S 3 = 3 S3. This implies (3.). This, combined with the monotonicity identity (1.3), implies r u = 3 B r (x ) S3 + y x u B r (x ) y x, r>small. (3.4) If B r (x ) u = 3 S3 r u holds for some r >, then (3.4) implies y x =fora.e. y B r (x ). Therefore u(x) =φ( x x x x )forx B r (x ), with some nontrivial smooth harmonic map φ C (S 3, S 3 ). Applying Theorem 3.1 again, we have that there is a Q O(3) such that u(x) =Q ( x x x x ) in Br (x ). Now we are ready to prove Theorem. Proof of Theorem Since u H 1 (B 4, S 3 ) is a stable harmonic map, the stablity inequality (1.6) implies u η 1 η, η C B 3 (B 1 4 ). (3.5) 4 B 4

7 Stable Stationary Harmonic Maps to Spheres 7 This implies, for any <ɛ<1, u Cɛ 1. (3.6) B 1 ɛ Therefore (1.7) follows from (3.6) and the monotonicity identity (1.3). To show (ii), we may assume for simplicity ɛ = 1 and prove that the number of singular points of u inside B 1 is uniformly bounded. This follows if we can prove that there exists a universal constant δ > such that the distance between any two singular points of u inside B 1 is at least δ. Suppose that this is false. Then we may assume that there exist a sequence of stable stationary harmonic maps u i H 1 (B 4, S 3 ) and a sequence of points x i B 1 with x i such that {,x i } S(u i ). Now we claim that there exists another universal constant δ 1 > such that B xi u i ( ) 3 S3 + δ 1 ( x i ). (3.7) For otherwise, the rescaled maps v i (x) =u i ( x i x) :B S 3 satisfy lim i v i = 3 B S3, so that the compactness theorem of [3] implies that there exist a stable stationary harmonic map v H 1 (B 4, S 3 )andapointx S 3 such that v i v in H 1 (B ), {,x } S(v ), and B v = 3 S3. This, combined with Proposition 3., implies v (x) =Q( x x )inb for some Q O(3). Hence S (v )={}. We get the desired contradiction. Proof of Theorem 3 With the help of Theorem 3.1, Proposition 3., the proof of Theorem 3 can be carried out exactly in the same manner as in Hardt Lin [1]. We omit it here. We end this section with a remark. Remark 3.3 For m, k 3, if u H 1 (B n, S k ) is a stable stationary harmonic map, then S (u), the singular set of u, isan(n ˆd(k) 1)-rectifiable set. In fact, since the set containing all stable stationary harmonic maps from B n to S k forms a compact subset of Hloc 1 (Bn, S k ) by the compactness theorem of [3], one can apply Simon s proof [1] to this case to yield the rectifiablity of S (u). 4 Harmonic Maps into S Since the stability inequality (1.6) for stable harmonic maps into S has no implication on the regularity, we consider a special class of harmonic maps that are obtained from minimizing sequences among continuous maps into S. Some general discussions have been presented by Lin [13], but the S case seems of particular interest. For n 3, asssume that g C ( B n, S ) has a continuous, H 1 -extension map G C (B n, S ) H 1 (B n, S ). Consider an energy minimizing sequence {u i } C H 1 (B n, S ), with u i B n = g. After taking possible subsequences, we may assume that u i u weakly in H 1 (B n, S ) and there exists a nonnegative Radon measure μ on B n such that u i dx μ as convergence of Radon measures. Moreover, by Fatou s lemma, we know that there is a nonnegative Radon measure ν, called a defect measure, such that μ = u dx + ν. Lemma 4.1 u H 1 (B n, S ) is a weakly harmonic map. Proof Let φ C(B 1 n, R 3 ), consider the maps u t i (x) =u i(x) t(φ(x) u i (x)) : B n S for t sufficiently small. Then it is easy to see u t i(x) =1+O(t ), u t i(x) = u i (x) +t (φ u i )(x), u i (x) + O(t ).

8 8 Lin F. H. and Wang C. Y. Setting wi t(x) = ut i (x) u t i (x) : Bn S,wehave lim wi t (x) lim u i (x). i B n i B n Note that wi(x) t = ut i (x) u t i (x) ut i (x),ut i (x) ut i (x) u t, i (x) 3 so that Bn wi(x) t u t i = (x) B u t ut i (x),ut i (x) n i (x) u t. i (x) 4 Since u t i (x),ut i (x) = O(t) weobtain lim wi(x) t = lim u t i B n i i(x) + O(t ) B n = lim u i +t (φ u i )(x), u i (x) + O(t ). i B n B n This implies lim i (φ u B n i )(x), u i (x) =. This implies Δu u =inthesenseof distributions. Since u = 1, this implies u satisfies Δu + u u =, and hence u is a weakly harmonic map. For the limit Radon measure μ, wehave Lemma 4. For any a B n, <r<d a =1 a, then μ(b r(a)) r is monotonically nondecreasing with respect to r. In particular, Θ n μ(b n (μ, a) = lim r (a)) r r exists for any a B n n and is uppersemicontinuous. Proof See [13] Lemma.. The basic reason is that a homogeneous of degree zero extension is essentially admissiable, with some replacement near the origin by suitable rescalings of the minimizing sequence. For the limit Radon measure μ and the limit map u, we have the following Caccioppolli type inequality. Lemma 4.3 There is an ɛ > such that for any θ (, 1 ) there exists a C θ > such that if μ(b r(a)) (r) n where u Br (a) = ɛ for B r (a) B n,then R B r (a) u B r (a). μ(b r (a) r n θ μ(b r(a)) (r) n + C θ B r (a) u u B r (a) Proof See [13] Lemma.4 and Lemma.6. Now we state two consequences of the Caccoppolli inequality (4.1). Proposition 4.4 then (r) n, (4.1) There exist ɛ > and C > such that if μ(b r(a)) (r) ɛ n for B r (a) B n μ(b r (a)) B r n C r (a) u u B r (a) (r) n. (4.) Proof (4.) follows from (4.1) and a covering argument (see Simon [14]). Proposition 4.5 There exists an ɛ > such that if μ(b r(a)) (r) n ɛ for B r (a) B n,then u C (B r (a), S ). Proof First observe that (4.) implies that if μ(b r(a)) (r) n ɛ then we have the reverse Hölder inequality: B r (a) u B r n C r (a) u u B r (a) (r) n. (4.3)

9 Stable Stationary Harmonic Maps to Spheres 9 It is well known that a weakly harmonic map u H 1 (B n, S ) satisfying (4.3) enjoys the partial regularity as above. The reader can also see [13] Theorem.3. Remark 4.6 A direct consequence of (4.) and Proposiition 4.6 is the following: There exist ɛ > andc 1 > such that if for B r (a) B n, μ(b r (a)) ɛ r n then μ(b r (a)) r n C 1 r, <r r. (4.4) In particular, there exists a nonnegative f L (B r (a)) such that ν = f(x) dx. (4.5) Proof Proposition 4.5 implies that u C (B r (a),s )and u C (B r (a)) C 1 r 1. (4.6) Therefore (4.) implies ( ) μ(b r (a)) r r n C 1, <r r r. (4.7) This implies (4.4). (4.5) follows from (4.4). In fact, we have Proposition 4.7 There exists an ɛ > such that if μ(b r (a)) ɛ r, then ν in B n r (a). Assuming for the moment the conclusion of Proposition 4.7, we have the following theorem. Theorem 4.8 Under the above notations, set Σ={a B n :Θ n (μ, a) ɛ }. Then (i) Σ is a closed subset and H n (Σ B r ) is finite for any <r<1. (ii) Σ is an (n )-dimensional rectifiable set. (iii) u C (B n \ Σ, S ). (iv) ν in B n \ Σ and u i u in Hloc 1 (Bn \ Σ, S ). Proof (i) follows from the Vitali s covering Lemma. (iii) follows from Proposition 4.5. (iv) follows from Proposition 4.7. (ii) has been proved by [13] Theorem 3.5 (see also Lin [15]). Now we return to the proof of Proposition 4.7. Proof of Proposition 4.7 For n =, 3, we can refer to the proof by [13, Lemma 3.]. For n 4, the original proof of [13, Lemma 3.] has a mistake, we outline a proof that is a suitable modification of that of Luckhaus extension Lemma (see [16]). Since the proof is based on inducation on n 4, for simplicity we only consider the n =4case. To prove f inb r, it suffices to show that for any a B r,thereexistsδ(r) >, with lim r δ(r) =, such that ν(b r (a)) δ(r)r n, <r< r 4. (4.8) By scalings, we may assume that a =andr =. From Proposition 4.5 and (4.6), we have u C (B, S )with u C (B 1 ) Cɛ. (4.9) By the Fubini s theorem, we may assume that {u i } C H 1 (B, S ) is the minimizing sequence such that u i u weakly in H 1 (B, S )and u i Cɛ, i 1, lim u i u =. (4.1) B i 1 B 1 Now we need to prove the following Lemma. Lemma 4.9 For any δ (, 1), there exists a sequence of maps v i C H 1 (B 1 \ B 1 δ, S ) such that ( ) v i B1 = u i,v i B1 δ = u, (4.11)

10 1 Lin F. H. and Wang C. Y. and ( ( ) v i Cδ u i + ) u B 1 \B 1 δ B 1 ( ) + Cδ 1 x u i(x) u +(i 1 ). (4.1) B 1 Assuming Lemma 4.9 for the moment, we can prove (4.8) as follows. Define another sequence of maps ū i C H 1 (B 1, S )by ( ) x u, x B 1 δ, ū i (x) = v i (x), x B 1 \ B 1 δ. Then we have u + ν(b 1 ) = lim u i lim ū i B i 1 B i 1 B 1 ) = lim (() n u + v i i B 1 B 1 \B 1 δ ( ) u + Cδ u + ν( B 1 ). B 1 B 1 This gives ν(b 1 ) Cδμ( B 1 ). Rescaling back to the original scales, we prove (4.8). Hence f inb r. The proof of Proposition 4.7 is complete. Proof of Lemma 4.9 As mentioned before, Lemma 4.9 has been proved by [13] Lemma 3. for n = 3. Now we want to show that Lemma 4.9 also holds for n =4. First we follow [16] to conclude that there exists a triangularization of S 3 by boxes of side lengths δ, {Δ i } L i=1 with L Cδ 3, such that L ( ( ) δ u i + ( ) u + ) i=1 Δ i u i u ( ( ) C u i + u + ) u i u. (4.13) S 3 By bi-lipschitz transformation, we may assume that each sub-annual region F j = {(r, θ) : r 1,θ Δ j } can be identified by G j =[1 δ, 1] Δ j for 1 j L. Observe that G j =({1} Δ j ) ({} Δ j ) ([, 1] Δ j. For 1 j L, applying Lemma 4.9 for n = 3, we conclude that there exists a map v i C H 1 ([, 1] Δ j, S ) such that ( ) v i {1} Δj = u i {1} Δj, v i {1 δ} Δj = u and ( ) v i Cδ u i + u + Cδ 1 [1 δ,1] Δ j Δ j Δ j {1 δ} Δ j ( ) u i u. (4.14) Now define w i C ( G j,s ) by letting w i = u i on {1} Δ j, w i = u( 1 δ )on{} Δ j, and w i = v i on [, 1] Δ j. We now divide it into two cases: (1) α =[w i ] Π 3 (S ) is trivial: In this case we know that w i has a continuous, H 1 - extension w i : G j S such that ( w i C w i G j G j ). (4.15)

11 Stable Stationary Harmonic Maps to Spheres 11 Therefore we can use the same construction as that in the proof of Lemma 4. (see also [13, Lemma.]) to find an essentially homogeneous of degree zero extension map w i C H 1 (G j, S )ofw i such that for any ɛ> ( ) w i Cδ w i +Ci 1 w i Cδ w i +i 1 C w i. (4.16) G j G j G j G j G j () α =[w i ] Π 3 (S ) is nontrivial: Although w i doesn t have continuous extension from G j to S, the theorem of White [17] implies that α 1 Π 3 (S ) can be represented by a map f i C 1 (S 3, S ) such that f i p i in S 3 \ B ɛi for some p i S,withɛ i, and f i i 1. (4.17) B ɛi Now we first modify w i on a small ball B ɛi [, 1] Δ j to obtain w i such that [ w i ]=[w i ]=α, w i p i in B ɛi [, 1] Δ j,and w i = w i +(ɛ i ). (4.18) G j G j Next we glue w i with f i along B ɛi [, 1] Δ j and denote the resulting map by ŵ i. It is readily seen that [ŵ i ] Π 3 (S )istrivial, ( ) ŵ i {1} Δj = u i {1} Δj, ŵ i {1 δ} Δj = u, {1 δ} Δ j and ŵ i w i + Ci 1 + O(ɛ i ). (4.19) G j G j Now we can follow the same construction as in (1) to obtain an extension w i C H 1 (G j,s ) such that w i = u i on {1} Δ j, w j = u( 1 δ )on{1 δ} Δ j,and w i Cδ w i + O(i 1,ɛ i ) G j G j [ ( ) Cδ u i + ] u + v i + O(i 1,ɛ i ) Δ j [1 δ,1] Δ j [ ( ) Cδ u i + ( ) u + δ u i + Δ j u Δ j ( ) + δ 1 ] u i u + O(i 1,ɛ i ). (4.) Δ j Finally we repeat the above construction over G j for 1 j L to obtain an extension map w i C H 1 (B 1 \ B 1 δ, S ) that satisfies (4.11) and L ( ) w i Cδ u i + u B 1 \B 1 δ i=1 Δ j L ( ( ) + Cδ δ u i + u i=1 Δ j ( ) + δ 1 ) u i u + O(i 1 ). [1 δ,1] Δ j This, combined with (4.13), implies (4.1). Therefore the proof of Lemma 4.9 is complete. References [1] Schoen, R., Uhlenbeck, K.: A regularity theory for harmonic maps. J. Differential Geom., 17(), (198)

12 1 Lin F. H. and Wang C. Y. [] Schoen, R., Uhlenbeck, K.: Regularity of minimizing harmonic maps into the sphere. Invent. Math., 78(1), 89 1 (1984) [3] Hong, M. C., Wang, C. Y.: On the singular set of stable-stationary harmonic maps. Calc. Var. Partial Differential Equations, 9(), (1999) [4] Price, P.: A monotonicity formula for Yang-Mills fields. Manuscripta Math., 43( 3), (1983) [5] Schoen, R.: Analytic aspects of the harmonic map problem, Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983), , Math. Sci. Res. Inst. Publ.,, Springer, New York, 1984 [6] Yau, S. T.: Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. Indiana Univ. Math. J., 5(7), (1976) [7] Nakajima, T.: Singular points for harmonic maps from 4-dimensional domains into 3-spheres. Duke Math. Journal., t appear [8] Hardt, R., Lin, F. H.: Stability of singularities of minimizing harmonic maps. J. Differential Geom., 9(1), (1989) [9] Almgren, F., Lieb, E.: Singularities of energy minimizing maps from the ball to the sphere: examples, counterexamples, and bounds. Ann. of Math. (), 18(3), (1988) [1] Hardt, R., Lin, F. H.: The singular set of an energy minimizing map from B 4 to S. Manuscripta Math., 69(3), (199) [11] Eells, J., Lemaire, L.: A report on harmonic maps. Bull. London Math. Soc., 1(1), 1 68 (1978) [1] Simon, L.: Rectifiability of the singular set of energy minimizing maps. Calc. Var. Partial Differential Equations, 3(1), 1 65 (1995) [13] Lin, F. H.: Mapping problems, fundamental groups and defect measures. Acta Mathematica Sinica, English Series, 15(1), 5 5 (1999) [14] Simon, L.: Singularities of geometric variational problems, Nonlinear partial differential equations in differential geometry (Park City, UT, 199), 185 3, IAS/Park City Math. Ser.,, Amer. Math. Soc., Providence, RI, 1996 [15] Lin, F. H.: Gradient estimates and blow-up analysis for stationary harmonic maps. Ann. of Math. (), 149(3), (1999) [16] Luckhaus, S.: Partial Hlder continuity for minima of certain energies among maps into a Riemannian manifold. Indiana Univ. Math. J., 37(), (1988) [17] White, B.: Homotopy classes in Sobolev spaces and existence of energy minimizing maps. Acta Math., 16, 1 17 (1984)