Annals of Global Analysis and Geometry 21: 141–149, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
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A Bridge Principle for Harmonic Diffeomorphisms between Surfaces YNGING LEE, AI NUNG WANG and DERCHYI WU Institute of Mathematics, Academia Sinica, 11529 Nankang, Taipei, Taiwan, R.O.C. e-mail:
[email protected] (Received: 31 October 2000) Abstract. We show a bridge principle for harmonic diffeomorphisms between closed surfaces with higher genus. Mathematics Subject Classification (2000): 53C99. Key words: bridge principle, energy, harmonic diffeomorphism, minimal submanifold.
1. Introduction Let 0 , 1 be two minimal surfaces with nonempty boundaries and with a thin strip connecting their boundaries. The bridge principle for minimal surfaces says that if the strip is thin enough, it should usually be possible to deform these minimal surfaces and the strip slightly to make a new minimal surface. This problem has been studied by many mathematicians [7, 8, 10, 11]. For harmonic maps of special kind (defining on domains of R n ), Mou [6] justified a bridge principle; whileas for harmonic maps on general submanifolds with boundaries, Lee et al. [4] proved a bridge principle. The technique of their proofs is similar to previous bridge constructions for minimal submanifold in Rn [8]; constructing an approximate solution on the connected sum by gluing together f0 and f1 , and then solving the harmonic map equation by linearization, elliptic estimates, and fixed point methods. In this paper, we provide a bridge principle for harmonic diffeomorphisms between closed surfaces with genus number larger than two. In fact, ˜ι THEOREM. Let ι be embedded closed surfaces of a Riemannian manifold M, ˜ ˜ be embedded closed surfaces of a Riemannian manifold M, dimM, dim M ≥ 3, ˜ ι ) ≥ 2, for ι = 0, 1. Let fι : ι → ˜ ι be harmonic and genus (ι ) = genus ( ˜ 0 and differmorphisms. Suppose that γ , γ˜ are Jordan arcs connecting 0 and 1 , ˜ ι = γ˜ (ι) = fι (γ (ι)), for ˜ 1 , respectively, satisfying that γ ∩ ι = γ (ι), and γ˜ ∩ ι = 0, 1. Then in any sufficiently small -neighborhood of γ (small -neighborhood ˜ 0 and ˜1 of γ˜ , respectively), one can connect 0 and 1 by a bridge tube T , (
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by a bridge tube T˜ , respectively), and find a harmonic diffeomorphism F from ˜ 0 T˜ ˜ 1. the smooth bridged surface 0 T 1 onto the smooth bridged surface Furthermore, as the radius of T shrinks to zero (i.e., → 0), {F } converges to fι (in the C k -topology, k > 2) on each compact subset of ι \γ , for ι = 0, 1. By a bridge tube on 0 ∪ 1 along γ , we say the image T of a diffeomorphism φ from [0, 1] × S 1 into M, such that φ = φ|[0,1]×∂D , φ: [0, 1] × D → M, with D the unit disk in R 2 , φ(r, 0) = γ (r), for 0 ≤ r ≤ 1, φ ∩ 0 = {φ(0, θ), θ ∈ S 1 }, and φ ∩ 1 = {φ(1, θ), θ ∈ S 1 }. If we ignore the centered line γ , then T is simply called a bridge tube joining 0 and 1 . Except for special indication, we will abuse the notions T (r, θ) or (r, θ) for φ(r, θ), with 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2π . Finally, 0 T 1 denotes the bridged surface (0 ∪ 1 ∪ T )\(φ(0, D) ∪ φ(1, D)). We remark that the main point of the theorem is the limit behavior of the harmonic diffeomorphisms F , as → 0. The key ingredients of the proof are the diffeomorphic and unique property of the harmonic maps involved. For surfaces with higher genus, the existence and uniqueness theorems for harmonic diffeomorphisms were proved by Jost and Schoen [3], and Coron and Helein [2]. This approach fails for general harmonic maps or arbitrary closed manifolds, since the existence and uniqueness theorems are absent. 2. Proof of the Theorem The proof of this theorem is decomposed into five steps. Step 1. Construction of the Approximate Harmonic Maps G Let zι = (r ι , θ ι ) be the geodesic coordinates [1] of a neighborhood centered at γ (ι) in ι (˜zι = (˜r ι , θ˜ ι ) the geodesic coordinates of a neighborhood centered at γ˜ (ι) in ˜ ι , respectively). We fix the orientations of zι , z˜ ι so that the normals of z0 , z˜ 0 are outward, and the normals of z1 , z˜ 1 are inward. For > 0, there exist 0 , 1 , λ˜ , a bridge tube T ⊂ M in the -neighborhood of γ , and a bridge tube T˜ ⊂ M˜ in the -neighborhood of γ˜ , satisfying that 0 T 1 , ˜ 1 are smooth closed surfaces, ˜ 0 T˜ T ∩ ι = ( ι , θ ι ), 0 ≤ θ ι ≤ 2π, in zι -coordinates, ˜ ι = (λ˜ , θ˜ ι ), 0 ≤ θ˜ ι ≤ 2π, in z˜ ι -coordinates, T˜ ∩ sup distance˜ ι γ˜ (ι), fι (( ι , θ)) = λ˜ , ι = 0, 1.
(1.1)
θ∈[0,2π]
By the diffeomorphic condition, we can have 0 ∼ 1 ∼ λ˜ ∼ , c0−1 f
(1.2)
≤ g ≤ c0 f with c0 a positive constant independent of . where f ∼ g iff Without loss of generality, the length of γ and γ˜ can be set to be one. Denote the induced metric tensors of T from M by µ (r, θ)2 dr 2 + 2τ (r, θ) dr dθ +
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˜ 2 d˜r 2 + 2τ˜ (˜r , θ˜ ) d˜r dθ˜ + ρ (r, θ)2 dθ 2 , 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2π , (µ˜ (˜r , θ) ρ˜ (˜r , θ˜ )2 dθ˜ 2 , 0 ≤ r˜ ≤ 1, 0 ≤ θ˜ ≤ 2π , the metric tensors of T˜ , respectively). By the calculation of the metric tensors of a small geodesic neighborhood centered at a point of γ (γ˜ , respectively) [1], and, if necessary, a smooth deformation of the interior of T (of T˜ respectively), we can assume these metric tensors satisfying that: ˜ ∼ 1, µ (r, θ), µ˜ (˜r , θ) ρ (r, θ), ρ˜ (˜r , θ˜ ) ∼ , ˜ = O( 3), |τ (r, θ)|, |τ˜ (˜r , θ)| |ρ˜ (˜r , θ˜ )/ρ˜ (˜r , θ˜ )| −→ 1, as → 0. unif
(1.3)
˜ θ˜ ≤ 2π , and f (s) = O(s k ) if and only if Here 0 ≤ r, r˜ , r˜ ≤ 1, and 0 ≤ θ, θ, there exists a constant c1 , such that f (s) ≤ c1 s k , as s → 0. ι ˜ι Let fιθ be the θ˜ ι -component of fι in zι -coordinates, and fιr˜ be the r˜ ι -component of fι in z˜ ι -coordinates, ι = 0, 1. By (1.1), (1.2), we can define the approximate ˜ 0 T˜ ˜ 1 } by harmonic maps {G : 0 T 1 → ˜ 0, f (z) ∈ 0 0 ˜0 ˜ 0, (f0r˜ (z)λ˜ , f0θ ) ∈ (r, H0, (r, θ)) ∈ T˜ , G (z) = (r, θ) ∈ T˜ , (r, H1, (r − 34 , θ)) ∈ T˜ , (f r˜ 1 (z) ˜ , f θ˜ 1 ) ∈ ˜ 1, λ 1 1 ˜ 1, f1 (z) ∈ where
r˜ ι
fι (z)λ˜ =
z ∈ 0 \(0 ≤ r 0 ≤
0 , 0 ≤ θ 0 ≤ 2π ), √ z = (r 0 , θ 0 ), 0 ≤ r 0 ≤ 0 , 0 ≤ θ 0 ≤ 2π, z = (r, θ) ∈ T , 0 ≤ r ≤ 14 , 0 ≤ θ ≤ 2π, z = (r, θ) ∈ T , 14 ≤ r ≤ 34 , 0 ≤ θ ≤ 2π, z = (r, θ) ∈ T , 34 ≤ r ≤ 1, 0 ≤ θ ≤ 2π, √ z = (r 1 , θ 1 ), 1 ≤ r 1 ≤ 1 , 0 ≤ θ 1 ≤ 2π, √ z ∈ 1 \(0 ≤ r 1 ≤ 1 , 0 ≤ θ 1 ≤ 2π ),
fιr˜ (z), if fιr˜ (z) ≥ λ˜ , ι if fιr˜ (z) < λ˜ , λ˜ , ι
√
ι
(1.4) ˜0
H0, : [0, 1/4] × S 1 → S 1 is a homotopy from f0θ ( 0 , ·) to the identity map on S 1 , and H1, : [0, 1/4] × S 1 → S 1 is a homotopy from the identity map on S 1 to ˜1 f1θ ( 1 , ·). Besides, for ι = 0, 1, and any fixed θ, Hι, (·, θ) is monotonic and of constant speed. Step 2. Energy Estimate of the Approximate Harmonic Maps G and Harmonic Maps F Denote e(G ) as the energy density of G , d(r ι )2 + (r ι )2 σι2 (r ι , θ ι )d(θ ι )2 the metric tensors of zι = (r ι , θ ι ) [1], d(˜r ι )2 + (˜r ι )2 σ˜ ι2 (˜r ι , θ˜ ι )d(θ˜ ι )2 , the metric tensors of
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z˜ ι = (˜r ι , θ˜ι ) [1], then the energy of G is e(G ) dv + E(G ) = √ 0 \(0≤r 0 ≤ 0 ,0≤θ 0 ≤2π)
1/4 2π
+
0 √ 1
+
0
2π
+
3/4 2π
1/4
0
0
1
+
3/4
0
2π
0
2π
e(G ) r 0 σ0 dr 0 dθ 0 +
0
e(G ) µ2 ρ 2 − τ 2 dr dθ +
e(G ) r 1 σ1 dr 1 dθ 1 +
0
1
√
√ 1 \(0≤r 1 ≤ 1 ,0≤θ 1 ≤2π)
e(G ) dv
= I + II + III + IV + III + II + I . The diffeomorphic property of f0 , (1.2) and (1.4) imply that f0r˜ (z)λ˜ ∼ √ 0 f0r˜ (z), for z = (r, θ), and 0 ≤ r ≤ 0 . In addition, note that σ˜ (r, θ) = 1+O(r 2 ) [1]. So there exists a uniform constant c, such that
0 0 0 0 √ 0 2π ∂f0r˜ λ˜ ∂f0r˜ λ˜ 1 ∂f0r˜ λ˜ ∂f0r˜ λ˜ 0 r σ0 + dr 0 dθ 0 + II = 0 0 0 0 0σ ∂r ∂r ∂θ ∂θ r 0 0 0 0
√
+
0
2π
0
0
˜0
{f0r˜ λ˜ σ˜ 0 (f0r˜ λ˜ , f0θ )}2 × 0
0
˜0 ˜0 ˜0 ∂f0θ ∂f0θ 1 ∂f0 ∂f0θ 0 r σ0 + dr 0 dθ 0 × ∂r 0 ∂r 0 ∂θ 0 ∂θ 0 r 0 σ0
√ 0 2π r˜ 0 r˜ 0 0 0 ∂f0r˜ ∂f0r˜ 1 ∂f0 ∂f0 0 ≤ r σ0 + dr 0 dθ 0 + 0 ∂r 0 0 ∂θ 0 r 0 σ ∂r ∂θ 0 0 0
θ˜ 0
+c ×
√ 0
2π
0
0 ˜0
˜0
{f0r˜ σ˜ 0 (f0r˜ , f0θ )}2 ×
˜0
0
0
˜0
˜0
∂f0θ ∂f0θ 1 ∂f0θ ∂f0θ 0 r σ + 0 ∂r 0 ∂r 0 ∂θ 0 ∂θ 0 r 0 σ0
dr 0 dθ 0
≤ cE(f0 |0≤r 0 ≤√ 0 ) = O( ). ˜0
Since H0, is a homotopy from f0θ ( 0 , ·) to the identity map, and for each fixed θ, H0, (·, θ) is monotonic and of constant speed. We obtain that ∂H0, ∂r ≤ 8π, θ ∂H0, ∂f0 ∂f0 sup ∂θ (r, θ) ≤ sup ∂θ ( 0 , θ) ≤ sup ∂θ ( 0 , θ) , 0≤θ≤2π,0≤r≤1/4 0≤θ≤2π 0≤θ≤2π
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and
2π
0
2 ∂H0, ∂θ (r, θ) dθ ≤ c , ˜0
for a constant c independent of r, . Here the onto property of f0θ ( 0 , ·) is used. Define
2
−1 2 µ τ ν * = . * + 2 τ ρ 2 By (1.3), we get ν ∼ 1, + ∼ −1 , and |* | = O( ). So 1/4 2π ν (r, θ)2 µ˜ (r, H0, (r, θ))2 µ2 ρ 2 − τ 4 dr dθ + III = 0
0
1/4
+
2π
ν (r, θ) ρ˜ (r, H0, (r, θ)) 2
0
2
0
∂H0, ∂r
2 µ2 ρ 2 − τ 4 dr dθ +
∂H0, 2 2 2 + (r, θ) ρ˜ (r, H0, (r, θ)) µ ρ − τ 4 dr dθ + + ∂θ 0 0 1/4 2π ∂H0, 2 2 2 ν (r, θ) τ˜ (r, H0, (r, θ)) µ ρ − τ 4 dr dθ + +2 ∂r 0 0 1/4 2π ∂H0, 2 2 * (r, θ)τ˜ (r, H0, (r, θ)) µ ρ − τ 4 dr dθ + +2 ∂θ 0 0 1/4 2π ∂H0, ∂H0, * (r, θ)ρ˜ (r, H0, (r, θ))2 µ2 ρ 2 − τ 4 dr dθ +2 ∂r ∂θ 0 0 ≤ c ,
1/4
2π
2
2
for a constant c. Similarly, we have IV ≤ c2 , and III + II ≤ c2 , for a uniform constant c2 . So E(G ) ≤ E(f0 ) + E(f1 ) + c3 . Here c3 is constant independent of . Note that G can be slightly smoothized to be a diffeomorphism with the same energy bound. Applying the existence theorem of Jost and Schoen [3], one can find a harmonic diffeomorphism F in the homotopy class of G with E(F ) ≤ E(f0 ) + E(f1 ) + c3 .
(2.1)
Step 3. Construction of the Auxiliary Maps F0, ˜ ι , with D˜ ι,λ˜ = N˜ ι, ,0 = First of all, let N˜ ι, ,a be the a-neighborhood of D˜ ι,λ˜ in {(˜r ι , θ˜ ι ), 0 ≤ r˜ ι ≤ λ˜ , 0 ≤ θ˜ ι ≤ 2π }, and let Dι, ι = {(r ι , θ ι ), 0 ≤ r ι ≤ ι , 0 ≤ θ ι ≤ 2π }. If for some r0 ∈ [0, 1], F (T (r0 , S 1 )) ⊂ N˜ 0, , √4 T˜ N˜ 1, , √4 ,
(3.1)
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then the diffeomorphic property of F implies that either for 0 ≤ r ≤ r0 , or for r0 ≤ r ≤ 1,
2 2π √ ∂F ≤ c4 dθ ∂θ ˜1 ˜ 0 ˜ 0 T
≤ ≤
c4 c4
2π
2 e(F )ρ (r, θ) dθ
0 2π
e(F )ρ (r, θ) dθ ·
0
2π
ρ (r, θ) dθ, 0
with c4 , c4 constants independent of . Integrating both sides of the above inequality with respect to 0 ≤ r ≤ r0 , or r0 ≤ r ≤ 1 and using (1.3), (2.1), we obtain that √ min{r0 , 1 − r0 } ≤ c4 . (3.2) Hence, (3.1), (3.2) imply that there exists a constant c5 independent of , such that √ √ (3.3) F (T (r, S 1 )) ⊂ N˜ 0, , √4 T˜ N˜ 1, , √4 , for c5 ≤ r ≤ 1 − c5 . We conclude from (2.1) that there exists an infinite subsequence {F } →0 satisfying either (3.4) E F |(0 ∪T (0≤r≤c5√ ,S 1 ))\D0, 0 < E(f0 ) + c3 or
E F |(1 ∪T (1−c5 √ ≤r≤1,S 1))\D1, 1 < E(f1 ) + c3 .
(3.5)
Suppose that case (3.4)√holds. By (3.3), one can define the auxiliary maps F0, ˜ 0 \D˜ 0,λ˜ by from (0 ∪ T (0 ≤ r ≤ c5 , S 1 ))\D0, 0 to ˜ 0 \D˜ 0,λ˜ ), F , if x ∈ F −1 ( F0, = √ ˜ 0 \D˜ 0,λ˜ )), p0, ◦ F , if x ∈ (0 ∪ T (0 ≤ r ≤ c5 , S 1 ))\(D0, 0 ∪ F −1 ( with p0, : N˜ 0, , √4 T˜ N˜ 1, , √4 → ∂ N˜ 0, ,0 being the contraction map ˜ p0, (˜r , θ˜ ) = (0, θ), ˜ 0 T˜ ˜ 1 is parametrized by (˜r , θ˜ ), −a ≤ r˜ ≤ 1 + b, where N˜ 0, ,a T˜ N˜ 1, ,b ⊂ 0 ≤ θ˜ ≤ 2π , i.e. N˜ 0, ,a \D˜ 0,λ˜ be parametrized by (−a ≤ r˜ ≤ 0, 0 ≤ θ˜ ≤ 2π ), T˜ by (0 ≤ r˜ ≤ 1, 0 ≤ θ˜ ≤ 2π ), and N˜ 1, ,b \D˜ 1,λ˜ by (1 ≤ r˜ ≤ 1+b, 0 ≤ θ˜ ≤ 2π ).
Step 4. Energy Estimate of the Auxiliary Maps F0, Now we turn to estimating the upper bound of the energy of F0, . For simplicity
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and explicity, we write xι , xj for the local coordinates of the domains of F ; R, 2 ˜ 2) ˜ the local coordinates for the local coordinates for the domain of p0, ; and (R, the image of p0, . Also, the metrics are denoted by gij, dxi dxj , µ˜ (R, 2)2 dR 2 + ˜ 2) ˜ 2) ˜ 2 dR˜ 2 + 2τ˜ (R, ˜ dR˜ d2 ˜ + 2τ˜ (R, 2) dR d2 + ρ˜ (R, 2)2 d22 and µ˜ (R, 2 ˜2 ˜ ˜ ρ˜ (R, 2) d2 respectively. We obtain e(p0, ◦ F ) =
ij
≤
˜
˜ 2)) ˜ 2 )(ρ˜ (R, 2))2 g ij (x)((ρ˜ (R, 2))−2 (ρ˜ (R,
ij
≤ c
˜
∂p 2 ∂p 2 ˜ 2)) ˜ 2 0, 0, g ij (x) (ρ˜ (R, 2))−2 (ρ˜ (R, ∂2 ∂2
g ij (x)(ρ˜ )2 (R, 2)
ij
(ρ˜ (R, 2))2
∂F 2 ∂F 2 ∂xi ∂xj
∂F 2 ∂F 2 ∂xi ∂xj
∂F 2 ∂F 2 ∂xi ∂xj
≤ c e(F ), for √ ˜ 0 \N˜ 0, ,0 ) x ∈ ((0 ∪ T (0 ≤ r ≤ c5 , S 1 ))\D0, 0 )\F −1 ( and ˜ 2) ˜ 2 ρ˜ (R, . ˜ ˜ ρ˜ (R, 2)2 R,R,2, 2
c = max
By (1.3), one can have c → 1. Hence, lim E(F0, ) ≤ lim max{1, c }E(F |(0 \D0, o )∪T (0≤t ≤c5√ ,S 1 ) )
→0
→0
≤ lim max{1, c }(E(f0 ) + c3 ) →0
= E(f0 ),
(4.1)
by assuming that (3.4) holds. Step 5. Limit Behavior of F0, and F Denote DF0, as the domain of F0, , that is, √ DF0, = (0 ∪ T (0 ≤ r ≤ c5 , S 1 ))\D0, 0 . Since F is a diffeomorphic map and p0, is a contraction map. For any sufficiently small disk Dr (z), F0, (Dr (z)∩DF0, ) is the simply connected component of F0, (DF0, )\F0, (∂{Dr (z) ∩ DF0, }). Therefore, by the Courant–Lebesque lemma
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and (4.1), {F0, } (except for finitely many F0, ) is equicontinuous on any compact subset of 0 \γ (0) [3, lemma 1]. Furthermore, by the weak compactness of H21 , the Arzela–Ascoli theorem and a diagonal argument, we can show that ˜ 0 , and on each compact subset of 0 \γ (0), there exists a map F0 : 0 \γ (0) → 1 {F0, } tends to F0 in the weak H2 -topology and uniformly in the C 0 -topology. ˜ 0 \f0 (γ (0)) by the Note F0, (∂{DF0, }) = ∂{F0, (DF0, )} and the image of F0 is diffeomorphic property of F and (3.3). F0 can be continuously extended to be a ˜ 0 , denoted by F0 still. Furthermore, the lower semicontinuity map from 0 onto of the energy functional and (4.1) imply E(F0 ) ≤ E(f0 ). By [2], f0 is an energy minimizer. So F0 is a stationary harmonic map [9]. A regularity theorem of Morrey [5] (or [9, lemma 1.1]) then says F0 is a smooth harmonic map. Therefore, the uniqueness theorem for degree one energy minimizer between surfaces with higher genus [2] concludes that F0 = f0 . ˜ 0 \D˜ 0,λ˜ )). One can slightly Since F0, is smooth except on the curve ∂(F −1 ( smoothize F0, to obtain a map F0, so that is smooth, F0, ) ≤ E(F0, ) + , E(F0, = F0, , F0,
on
DF0, \O ,
˜ 0 \D˜ 0,λ˜ )) ≤ . distanceDF0, O , ∂(F −1 (
(5.1)
The convergence of {F0, } to f0 and (5.1) and (3.3) imply that on each compact tends to f0 in the C k -topology, k > 2. subset of 0 \γ (0), F0, Furthermore, we claim that for any α > 0, there exists an α , such that ˜ 0 \D˜ 0,λ˜ ) ∪ O , 0 \D0,α ⊂ F −1 (
for ≤ α .
(5.2)
Suppose the claim is not true, we then can find x ∈ 0 \D0,α , such that F (x ) ∈ N˜ 0, ,0 T N˜ 1, , √4 \O . Hence the definition of F0, and (5.1) imply that Jacobian of vanishes at x . Taking a limit of {x }, we can find a point xα ∈ 0 \D0,α , such F0, that the Jacobian of f0 vanishes at xα . This contradicts with the diffeomorphic property of f0 . So (5.2) has been justified. conclude that on The definition of F0, , (5.1), (5.2) and the convergence of F0, k each compact subset of 0 \γ (0), {F } converges to f0 in the C -topology, k > 2. Now (3.5) holds with a possible larger constant c3 . The same argument can be used to prove {F } C k -converging to f1 on compact subsets of 1 \γ (1). References 1. 2. 3.
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Lee, I. Y., Wang, A. N., and Wu, D. C.: A bridge principle for harmonic maps, Ann. Global Anal. Geom. 18 (2000), 107–127. Morrey, C. B.: Multiple Integrals in the Calculus of Variations Grundlehren 130, SpringerVerlag, Berlin, 1966. Mou, L.: Boundary value problems for harmonic and minimal maps of surfaces into manifolds, Comm. Partial Differential Equations 14 (1989), 1509–1540. Meeks, W. H. and Yau, S. T.: The existence of embedded minimal surfaces and the problem of uniqueness Math. Z. 179 (1982), 151–168. Smale, N.: A bridge principle for minimal and constant mean curvature submanifolds of R N , Invent. Math. 90 (1987), 505–549. Schoen, R.: Analytic aspects of the harmonic map problem, in: Seminar on Nonlinear PDE, MSRI Publication, Springer, New York, 1984, pp. 321–358. White, B.: The bridge principle for stable minimal surfaces, Calc. Var. Partial Differential Equations 2 (1994), 405–425. White, B.: The bridge principle for unstable and for singular minimal surfaces, Comm. Anal. Geom. 2(4) (1994), 513–532.
