Mathematische Annalen

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1 Math. Ann. 319, (2001) Digital Object Identifier (DOI) /s Mathematische Annalen A Moebius characterization of Veronese surfaces in S n Haizhong Li Changping Wang Faen Wu Received August 12, 1999 / Published online March 12, 2001 Springer-Verlag 2001 Abstract. Let M m be an umbilic-free submanifold in S n with I and II as the first and second fundamental forms. An important Moebius invariant for M m in Moebius differential geometry is the so-called Moebius form Φ, defined by Φ = ρ 2 i, {H,i + j (II ij H I ij )e j (log ρ)}ω i e, where {e i } is a local basis of the tangent bundle with dual basis {ω i }, {e } is a local basis of the normal bundle, H = H e is the mean curvature vector and ρ = m m 1 II HI.In this paper we prove that if x : S 2 S n is an umbilics-free immersion of 2-sphere with vanishing Moebius form Φ, then there exists a Moebius transformation τ : S n S n and a 2k-equator S 2k S n with 2 k [n/2] such that τ x : S 2 S 2k is the Veronese surface. Mathematics Subject Classification (2000): 53A30, 53C42, 53A Introduction Let M m be an umbilic-free submanifold in S n with I and II as the first and second fundamental forms. Then the euclidean invariant Φ = ρ 2 i, {H,i + j (II ij H I ij )e j (log ρ)}ω i e is also invariant under the Moebius transformation group in S n, called Moebius form of M m, where {e i } is a local basis of the tangent bundle with dual basis {ω i }, {e } is a local basis of the normal bundle, H = H e is the mean curvature vector and ρ = m II HI. This Moebius invariant plays an important role m 1 H. Li Department of Mathematical Sciences, Tsinghua Unviersity, Beijing , People s Republic of China ( hli@math.tsinghua.edu.cn) C.P. Wang Department of Mathematics, Peking University, Beijing , People s Republic of China ( wangcp@pku.edu.cn) F. Wu Department of Mathematics, Northern Jiaotong University, Beijing, , People s Republic of China ( fewu@center.njtu.edu.cn) The first author is supported by the project No of NSFC and the second author is supported by 973 Project, RFDP, Quishi Award and DFG466-CHV-II3/127/0.

2 708 H. Li et al. in Moebius differential geometry. It appears naturally in the study of Moebius isoparametric hypersurfaces and Willmore surfaces in S n. Let x : M S n be a surface. The Willmore functional for surfaces in S n is defined by W(x) = ( II 2 2 H 2 )dm, M where dm is the volume form for M. It is known that W is a Moebius invariant. If we decompose the Moebius invariant Φ into its (1,0)-part and its (0,1)-part with respect to the complex structure of M, Φ = (φ dz + φ d z) e, then the Euler-Lagrange equations for the Willmore functional W can be written as (φ ) z + β φ β Ā β 1 2 e 2ω ψω = 0, here Ψ := ψdz 2, Ω := Ω dz 2 E and the the normal connection θ β := A β dz+a β d z are global Moebius invariants on M (cf. (1.12), (1.13) and (1.16) in Sect. 1). The above equations show that the Moebius invariant Φ is important for the study of Willmore surfaces in S n. In this paper we use Φ to characterize the Veronese surfaces in S n. Our main result is the following Main Theorem. Let x : S 2 S n be an umbilic-free immersion of 2-sphere with vanishing Moebius form Φ,then there exists a Moebius transformation τ : S n S n and a 2k-equator S 2k S n with 2 k [n/2] such that τ x : S 2 S 2k is the Veronese surface. This paper is organized as follows. We will give Moebius invariants and structure equations for surfaces in S n in Sect. 1 and prove the main theorem in Sect Moebius invariants for surfaces in S n Let R n+2 1 be the Lorentz space R n+2 with the inner product <, > given by <X,X>:= x 0 y 0 + x 1 y 1 + +x n+1 y n+1, (1.1) where X = (x 0,x 1,,x n+1 ), Y = (y 0,y 1,,y n+1 ) R n+2 1. We denote by C+ n+1 the half cone in R n+2 1 and by Q n the quadric in RP n+1 C n+1 + := {X R n+2 <X,X>= 0,x 0 = p(x) > 0}, (1.2) Q n := {[X] RP n+1 <X,X>= 0}, (1.3) where p : R n+1 1 R is the projection of X to its first coordinate x 0. Let O + (n + 1, 1) be the Lorentz group of R n+2 1 preserving the inner product <, >

3 A Moebius characterization of Veronese surfaces in S n 709 and C+ n+1 invariant. Then O + (n + 1, 1) is a transformation group of Q n defined by T([X]) := [XT ], X C+ n+1, T O+ (n + 1, 1). A classical theorem states that (see [5]) Theorem 1.1 Two submanifolds x, x : M S n are Moebius equivalent if and only if there exists T O + (n + 1, 1) such that [1, x] =T([1,x]) : M Q n. Let x : M m S n be a submanifold. Let y : U C n+1 + be a local lift of [1,x]:M m Q n, defined on an open set U of M m. We denote by and κ the Laplace operator and the normalized scalar curvature with respect to the metric <dy,dy>, then g = (< y, y > m 2 κ)<dy,dy> (1.4) is independent of the local lift y and thus globally defined on M m (see [5]). Moreover, g is positive definite at any non-umbilic point of M m. It is clear from Theorem 1.1 that g is a Moebius invariant. We call it Moebius metric of M m. Under the assumption that M m is umbilic point free we can find a unique lift Y : M m C+ n+1 such that g =< dy,dy >. Y is called canonical lift of [1,x]:M m Q n. It follows from Theorem 1.1 that Theorem 1.2 Two submanifolds x, x : M S n are Moebius equivalent if and only if there exists T O + (n + 1, 1) such that their canonical lift Y and Ỹ satisfy Y = ỸT : M m C+ n+1. Let Y be the canonical lift of M m. We define N : M m R n+2 1 by N = 1 m Y 1 < Y, Y >Y, (1.5) 2m2 where is the Laplace operator of the Moebius metric g, then we have <Y,Y >=<N,N>= 0, <Y,N>= 1, <N,dY>=<dN,Y >= 0. (1.6) Now let {e i } be a local orthonormal basis of Moebius metric. We have the following Lorentz orthogonal decomposition R n+2 1 = span{y, N} span{e 1 (Y ),,e m (Y )} V. We call V the Moebius normal bundle of M m. Let {E } be a orthonormal basis of V with respect to the Lorentz metric in R n+2 1, then we have a Moebius moving frame {Y, N, e i (Y ), E } in R n+2 1 for x : M m S n. Using this Moebius frame we can give structure equations and Moebius invariants for submanifolds in S n. For detail we refer to [5].

4 710 H. Li et al. In this paper we assume that x : M S n is a connected surface without umbilical points. Let z = u+iv be a local complex coordinate on M with respect to g, we can write g = 1 2 e2ω (dz d z + d z dz) (1.7) for some locally defined smooth function ω. From (1.7) and the fact g = <dy,dy >we get <Y z,y z >=< Y z,y z >= 0, < Y z,y z >= 1 2 e2ω. (1.8) We denote by the Laplacian of g and by K the Gaussian curvature of g, then we have by (1.7) Y = 4e 2ω Y z z, K = 4e 2ω ω z z. (1.9) It follows from (1.4) that < Y, Y >= 1 + 4K. (1.10) Since {Y, N, Y z,y z,e } is a moving frame in R n+2 1, for any vector W R n+2 1 we have W = <W,N>Y+ <W,Y >N+ 2e 2ω <W,Y z >Y z +2e 2ω <W,Y z >Y z + n 2 <W,E >E. =1 (1.11) We define ψ = 2 <N z,y z >, φ =<N z,e >, (1.12) Ω = 2 <Y zz,e >, A β =<(E ) z,e β >= A β. (1.13) We know that ψdz 2, Φ c = φ dz E and Ω = Ω dz 2 E are globally defined Moebius invariants. Using the formula (1.11) and equations (1.6) (1.13) we can write the structure equations with respect to the local

5 A Moebius characterization of Veronese surfaces in S n 711 frame {Y, N,Y z,y z,e 1,,E n 2 } as follows Y z = Y z, Y z = Y z N z = 1 8 (1 + 4K)Y z + e 2ω ψy z + φ E, N z = 1 8 (1 + 4K)Y z + e 2ω ψy z + Y zz = 1 2 ψy + 2ω zy z + 1 Ω E, 2 Y z z = 1 16 e2ω (1 + 4K)Y 1 2 e2ω N, Y z z = 1 2 ψy + 2ω z Y z + 1 Ω E, 2 (E ) z = φ Y e 2ω Ω Y z + β φ E, A β E β, (1.14) (E ) z = φ Y e 2ω Ω Y z + β Ā β E β. Using the identities N z z = N zz, Y zz z = Y z zz and (E ) z z = (E ) zz we get the following integrability conditions for the linear PDE system (1.14): ψ z = 1 2 e2ω K z Ω φ, (1.15) (φ ) z 1 2 e 2ω ψω + β φ β Ā β = ( φ ) z 1 2 e 2ω ψ Ω + β φ β A β,(1.16) e 4ω (Ω ) z = β Ω 2 = 1 4, (1.17) Ω β Ā β e 2ω φ, (1.18) (A β ) z (Ā β ) z = 1 2 e 2ω (Ω Ω β Ω Ω β )+ γ (Ā γ A γβ A γ Ā γβ ). (1.19) According to [5], we define Moebius form Φ = i C i ω i E, where Ci = ρ 2 {H,i + j (II ij H δ ij )e j (log ρ)}. It can be reformulated as Φ = <dn,e >E. Since dn = N z dz+ N z d z, we get in the surface case that Φ = (φ dz + φ d z) E. Thus x : M S n has vanishing Moebius form if and only if φ =<N z,e >= 0 for all.

6 712 H. Li et al. 2. The proof of main theorem In this section we give proof of the main theorem mentioned in Sect. 0. Let x : S 2 S n be a topological 2-sphere in the n-dimensional unit sphere with vanishing Moebius form, i.e., Then from (1.15) (1.18) we obtain φ = 0, 1 n 2. (2.1) ψ z = 1 2 e2ω K z, (2.2) ψω = ψ Ω, (2.3) e 4ω Ω 2 = 1 4, (2.4) (Ω ) z = β Ω β Ā β. (2.5) It follows from (2.5) that ( Ω 2 ) z = 0, (2.6) thus Ω 2dz4 is a globally defined holomorphic 4-form on S 2. Since any holomorphic form on S 2 vanishes identically, we get = 0. (2.7) Ω 2 We claim that the globally defined 2-form Ψ = ψdz 2 vanishes identically. Suppose that there is a point p S 2 with ψ(p) = 0, then in a neighbourhood of p on S 2 we can write ψ = ψ e iθ. It follows from (2.3) that Ω = Ω e 2iθ. Thus we get from (2.7) and (2.4) that 0 = Ω 2 = e2iθ Ω 2 = 1 4 e4ω+2iθ, a contradiction. We conclude that ψ 0. It follows from (2.2) that K = constant. Thus (S 2,g) is a sphere of constant curvature K > 0. From the second equation of (1.14) we obtain N z = 1 8 (1 + 4K)Y z, (2.8)

7 A Moebius characterization of Veronese surfaces in S n 713 thus we can find a constant vector c R n+2 1 such that c = N 1 (1 + 4K)Y. (2.9) 8 Since < N,N >=< Y,Y >= 0, < Y,N >= 1 and the first coordinate Y 0 = p(y) of Y is positive, then the first coordinate N 0 = p(n) of N is negative, thus we get from (2.9) that p(c) <0, <c, c >= 1 (1 + 4K). (2.10) 4 We write σ := K >0, < c, c >= σ 2. (2.11) Then we can find some T O + (n + 1, 1) such that ct = ( σ, 0,, 0). (2.12) Now let T = ( ) wv, w R. ub We define x : S 2 S n by x = v + xb w + x u. (2.13) Then we have [1, x] = [(1,x)T] with T O + (n + 1, 1). It follows from Theorem 1.1 that x is Moebius equivalent to x. It is easy to see that Ỹ := YT is the canonical lift of x and the map Ñ of x defined by (1.5) is given by Ñ = NT. Thus we get from (2.9) and (2.12) that ( σ, 0,, 0) = Ñ 1 2 σ 2 Ỹ. (2.14) We write Ỹ = ρ(1, x). Since < Ỹ,Ñ>= 1 and < Ỹ,Ỹ >= 0, we get from (2.14) that ρσ = 1. Therefore, ρ = 2/ 1 + 4K. Since g =< dy,dy >=< dỹ,dỹ>= ρ 2 d x d x, we know that the Gauss curvature K of the immersion x : S 2 S n is given by K = 4K 4K + 1. Moreover, we get from (1.5) and (1.10) that Ñ = 1 2 Ỹ 1 (1 + 4K)Ỹ. (2.15) 8

8 714 H. Li et al. If we denote by the Laplace operator of d x d x, then we have = ρ 2.It follows from (2.14) and (2.15) that x + 2 x = 0. Thus by a well-known theorem of Takahashi [3] we know that x : S 2 S n is a minimal sphere with constant curvature K. It follows from the result of Bryant [1], Calabi [2] or Wallach [4] that x is contained in some 2k-equator S 2k of S n with 2 k [n/2] as Veronese surface. Thus we complete the proof of the main Theorem. References 1. Bryant R., Minimal surfaces of constant curvature in S n, Trans.Amer. Math. Soc, 290 (1985), Calabi E., Minimal immersions of surfaces in Euclidean spheres, J. Differential Geom, 1 (1967), Takahashi T., Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 18 (1966), Wallach N., Extension of locally defined minimal immersions of spheres into spheres, Arch. Math, 21 (1970), Wang C.P., Moebius geometry of submanifolds in S n, Manuscripta Math, 96 (1998),

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