A theorem of H. Hopf and the Cauchy-Riemann inequality II

“main” — 2007/11/30 — 17:14 — page 525 — #1 A theorem of H. Hopf and the Cauchy-Riemann inequality II Hilário Alencar, Manfredo do Carmo, Isabel Fernández∗ and Renato Tribuzy Abstract. This is a sequel to “A theorem of H. Hopf and the Cauchy-Riemann inequality” [AdCT]. Here the result of the previous paper is extended (see the precise statement in Section 1 of the present paper) to surfaces in three-dimensional homogeneous Riemannian manifolds whose group of isometries has dimension four and the bundle curvature is nonzero, whereas in the previous paper only the case of vanishing bundle curvature was treated. Keywords: mean curvature, genus zero surface, Hopf’s quadratic form, Cauchy- Riemann inequality. Mathematical subject classification: Primary: 53C42; Secondary: 20A10. 1 Introduction Let E3 be a 3-dimensional simply-connected homogeneous riemannian manifold with a 4-dimensional isometry group. Such a manifold is a riemannian fibration with bundle curvature τ over a 2-dimensional space form with sectional curvature k. They are classified, up to isometries, by k and τ , where k and them τ by are 3 any k two . real When numbers 0 with (hence k 6k= 4τ 0 2 ), and we k2 hav +e τ 2 the 6= 0. spaces We M will 2 k denote . E ( , τ ) τ = 6= ( ) × R for When k τ 6= 0, 0, and we the have univ the ersal Ber co ger vering spheres, of the for k Lie > 0, group the SL Heisenber , for g k 3-space, 0. The =fibers of 3 k are geodesics (straight lines in the case 2(R) 0) the < trans- E ( , τ ) τ = lations along which generate a Killing vector field ξ. The bundle curvature τ is given by ∇X ξ = τ ξ × X, where ∇ is the covariant derivative, X is any vector R ∗Partially supported by MEC-FEDER Grant No. MTM 2004-00160. “main” — 2007/11/30 — 17:14 — page 526 — #2 field, and ×is the cross product in E3(k, τ). We refer to [D] for further details on these spaces and their immersed surfaces. For the case τ=0, Abresch and Rosenberg [AR1] considered a surface M immersed in M2(k)×Rand introduced a quadratic form on Mby Q(X,Y)=2Hα(X,Y)−khξ, Xihξ, Yi, where Xand Yare tangent vectors, αis the second fundamental form and H is the mean curvature of M. Introduce in Misothermal parameters (u, v) (this means that the induced metric on Mis ds2=λ2(du2+dv2)) and let z=u+iv be the corresponding complex parameter. Abresch and Rosenberg proved the (2,0)-component Q(2,0)of Qis holomorphic if H=const. This generalizes a previous result of H. Hopf [H] for the case where Mis immersed in R3. Furthermore, they showed that if Mis homeomorphic to a sphere, then Mis an embedded surface invariant by rotations of M2(k)×Raround the factor R. Alencar, do Carmo and Tribuzy [AdCT] showed that it is not necessary to assume that H=const. in the above result. They proved the following result Theorem A. Let Mbe a compact immersed surface of genus zero in M2(k)×R. Assume that |d H| ≤ g|Q(2,0)|, where |d H|is the norm of the differential d H of the mean curvature Hof M, and gis a continuous, nonnegative real function. Then Q(2,0)is identically zero, hence by [AR1],Mis a CMC (constant mean curvature) embedded surface invariant by rotations in M2(k)×R. The goal of the present paper is to generalize Theorem A for immersed surfaces Min E3(k, τ), for τ6= 0. For that, we first introduce the quadratic form Q(X, Y ) = 2(H + iτ )α(X, Y ) − (k − 4τ 2)hξ, Xihξ, Y i for tangent vectors X, Y in M. The expression of the above form was inspired in a joint paper of Abresch and Rosenberg [AR2] in which they announce that the theorem of [AR1] (case H = const.) can be extended to surfaces immersed in E3(k, τ ). As usual, we denote by Q(2,0) the (2, 0)-component of Q in the complex structure of M determined by the induced metric. We prove Theorem 1. Let M be a compact surface of genus zero immersed in E3(k, τ ) with mean curvature H. Assume that |d H| ≤ g|Q(2,0)|, “main” — 2007/11/30 — 17:14 — page 527 — #3 wheregis a continuous nonnegative real function. Then Q(2,0)is identically zero and, by [AR2],Mis a CMC surface invariant by rotations in E3(k,τ). 2 Preliminaries Set θ=H+iτ,c=k−4τ2and write Qas Q(X,Y)=2θ α(X,Y)−chξ, Xihξ, Yi. The (2,0)-component of Qis Q(2,0)=ψ(z)dzdz. Here z=u+iv, where (u, v) are isothermal parameters in M, i.e., ∂ ∂u,∂ ∂u=∂ ∂v ,∂ ∂v =λ2,∂ ∂u,∂ ∂v =0, and dz =1 √2(du +idv), dˉz=1 √2(du −idv). Set Z=1 √2∂ ∂u−i∂ ∂v ,Z=1 √2∂ ∂u+i∂ ∂v , so that dz(Z)=1=dˉz(Z), dz(Z)=dˉz(Z)=0. Notice also that Q(Z,Z)=ψ(z)and hZ,Zi = λ2,hZ,Zi = hZ,Zi = 0. We first prove the lemma below that will be used repeatedly in this section. Lemma 1. With the above notation ∇ZZ= ∇ZZ=0, where ∇is the covariant derivative in M. Proof. From the symmetry of the conexion we have ∇ZZ−∇ZZ=[Z,Z]. “main” — 2007/11/30 — 17:14 — page 528 — #4 Astraightforward computation shows that, for any function fon M, [Z,Z](f)=0, so that ∇ZZ=∇ZZ.Now set ∇ZZ=aZ +bZ = ∇ZZ. Then, since hZ,Zi = 0, 0=1 2ZhZ,Zi = h∇ZZ,Zi = bλ2. It follows that b=0 and ∇ZZ=aZ. Similarly, since hZ,Zi=0, 0=1 2ZhZ,Zi= h∇ZZ,Zi=aλ2 so that a=0 and ∇ZZ=∇ZZ=0. Next, we compute dψ dˉz=ZQ(Z,Z)=Z(2θhSZ,Zi−chξ, Zi2), where Sis the shape operator corresponding to α. Proposition 1. ZQ(Z,Z)=2Z(H)α(Z,Z)+2θλ2Z(H). Proof. ZQ(Z,Z)=2Z(θ)hSZ,Zi+2θh∇Z(SZ), Zi−2chξ, Zih∇Zξ,Zi −2chξ, Zihξ, ∇ZZi,(1) where we have used that ∇ZZ=0. In the Lemmas below, we compute separately h∇Z(SZ), Zi,h∇Zξ,Ziand hξ,∇ZZi. Lemma 2. h∇Z(SZ), Zi = h∇Z(SZ), Zi+cλ2hξ,Nihξ, Zi. Proof. By using that ∇ZZ=0 and Codazzi equation, we obtain ∇Z(SZ)=(∇ZS)(Z)+S(∇ZZ) =(∇ZS)(Z)+e R(Z,Z)N =(∇ZS)Z+chN,ξi(hZ, ξiZ−hZ,ξiZ), where e Ris the curvature of E3(k, τ) and we have used Corollary 3.2 of Daniel [D]. Finally, since hZ,Zi = 0, we conclude that h∇Z(SZ), Zi = h∇Z(SZ), Zi+chN, ξihZ, ξiλ2. “main” — 2007/11/30 — 17:14 — page 529 — #5 Lemma 3. h∇Zξ,Zi = iτ λ2hξ,Ni, where Nis the normal to the surface M. Proof. By using ([D], proof of the Proposition 3.2) we obtain ∇Zξ=τξ×Z=τ(hJZ, ξiN−hξ, NiJZ), where Jis the complex multiplication by i. Thus h∇Zξ, Zi=−τhξ, Niiλ2. Lemma 4. hξ,∇ZZi=λ2Hhξ,Ni. Proof.∇ZZ= ∇ZZ+α(Z,Z)N. Since ∇ZZ=0and α(Z,Z)=λ2H.(See the proof of Lemma 2 in [AdCT]), we conclude that hξ,∇ZZi=λ2Hhξ,Ni. Putting Lemmas 2, 3 and 4 in the expression (1) of ZQ(Z,Z), we obtain ZQ(Z,Z)=2Z(H)α(Z,Z)+2θh∇Z(SZ), Zi+2θcλ2hN, ξihZ, ξi −2ci τ λ2hξ,Zihξ, Ni−2cλ2Hhξ,Nihξ, Zi. Since θ=H+iτ, the two last terms of the above sum cancel out with the third term. Thus ˉ Z Q(Z,Z)=2Z(H)α(Z,Z)+2θh∇Z(SZ), Zi.(2) To conclude the proof of Proposition 1, we still need some information on the term h∇Z(SZ), Zi. Lemma 5. h∇Z(SZ), Zi = λ2Z(H). Proof. We first claim that ∇ZZ=Z(λ2) λ2Z. To see that, set ∇ZZ=aZ+bZ. Then h∇ZZ,Zi=bλ2=1 2ZhZ,Zi=0. “main” — 2007/11/30 — 17:14 — page 530 — #6 It follows that b=0, hence ∇ZZ=aZ. But h∇ZZ,Zi = ZhZ,Zi = Z(λ2)=ahZ,Zi. It follows that a=Z(λ2) λ2, and this proves our claim. Now, notice that Z(λ2H)=Z(hSZ,Zi)= h∇Z(SZ), Zi+hSZ,∇ZZi. Thus h∇Z(SZ), Zi = Z(λ2)H+λ2Z(H)−hSZ,∇ZZi =Z(λ2)H+λ2Z(H)−hSZ,ZiZ(λ2) λ2 =λ2Z(H), since hSZ,Zi = λ2H. By using Lemma 5 in Equation (2) we conclude the proof of Proposition 1. Proof of Theorem 1. By Proposition 1, we have |ZQ(Z,Z)|=|2Z(H)α(Z,Z)+2θλ2Z(H)| ≤|d H||λ||2α(Z,Z)+2θλ2|, where we have used that |Z(H)|= |d H(Z)|≤ |d H||Z|and |Z(H)|≤ |d H||λ|. By hypothesis, |d H| ≤ g|Q(2,0)|, for a continuous g≥0. Thus  dψ dˉz= |ZQ(Z,Z)| ≤ h|Q(2,0)|=h|ψ(z)|,(3) where h=g|λ||α(Z,Z)|+2λ2|H+iτ|, i.e., his a continuous nonnegative function on M. We now use the Main Lemma of [AdCT] which states that if a function ψ(z)satisfies the inequality (3) in a neighborhood Uof a zero z0of ψ, then either ψ≡0 in a neighborhood V⊂U of z0or, for all zin V, ψ(z)=(z−z0)mfm(z), m≥1,fm(z0)6= 0.(4) “main” — 2007/11/30 — 17:14 — page 531 — #7 From that we can conclude the proof of Theorem 1 in the same way as in ([AdCT] end of section 2). For completeness, we summarize the argument here. By the Main Lemma, either Q(2,0)≡0, hence by [AR2], Mis a surface of revolution, or Q(2,0)has a finite number of zeroes. We show that this case leads to a contradiction. Indeed, the equation Im[Q(Z,Z)dz2] = 0 gives rise to two fields of directions on Mwhose singularities are the zeroes of Q(Z,Z). The index of any of these fields at the singular point is equal to −m/2, where m is the order of the zero that appears in Equation (4). Since Mhas genus zero, the sum of the indices of this field of directions, by Poincaré Theorem ([H], Chapter III, sect. 2), is two. This is a contradiction, thereby concluding the proof of Theorem 1. Acknowledgments. Thisworkwasdonewhen I. Fernández wasvisiting IMPA in the second semester of 2005; she wishes to thank people at IMPA for making her stay a pleasurable experience. References [AR1] U. Abresch and H. Rosenberg, A Hopf differential for constant mean curvature surface in S2×Rand H2×R.Acta Math. 193 (2004), 141–147. [AR2] U. Abresch and H. Rosenberg, Generelized Hopf differentials. Matemática Contemporânea, 28 (2005). Sociedade Brasileira de Matemática, 1–28. [AdCT] H. Alencar, M. do Carmo and R. Tribuzy, A theorem of Hopf and the Cauchy- Riemann inequality. Preprint, IMPA, 2005. To appear in Comm. in Analysis and Geometry. [D] B. Daniel, Isometric immersions into 3-dimensional, homogeneous manifolds. Preprint, IMPA, 2005. To appear in Comm. Math. Helv. [H] H.Hopf, Differential Geometry in the Large.Lectures Notesin Math., Springer- Verlag, volume 1000, 1983. “main” — 2007/11/30 — 17:14 — page 532 — #8 Hilário Alencar Universidade Federal de Alagoas Instituto de Matemática 57072-900 Maceió, AL BRASIL E-mail: [email protected] Manfredo do Carmo IMPA, Estrada Dona Castorina, 110 22460-320 Rio de Janeiro, RJ BRASIL E-mail: [email protected] Isabel Fernández Depto de Matemática Aplicada I E.T.S. de Ingenieria Informatica Universidad de Sevilla Av. Reina Mercedes s/n 41012 – Sevilla SPAIN E-mail: [email protected] Renato Tribuzy Universidade Federal do Amazonas Instituto de Ciências Exatas Departamento de Matemática Av. Rodrigo Otávio Jordão Ramos, 3000 69077-000 Manaus, AM BRASIL E-mail: [email protected]