On conformal minimal immersions ofS 2 into ?P n

Bolton, J Stuart; Jensen, Gary R.; Rigoli, Marco; Woodward, L. M.

doi:10.1007/bf01458531

Cited by 193 publications

(251 citation statements)

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“…We are now in the position to give a variation of the arguments used in the harmonic sequence construction [18], [2]. Eventually, this will give a characterization of Willmore surfaces under some assumptions on their normal bundle degrees.…”

Section: The Willmore Sequencementioning

confidence: 99%

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Sequences of Willmore surfaces

Leschke

Pedit

2007

Math. Z.

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Abstract. In this paper we develop the theory of Willmore sequences for Willmore surfaces in the 4-sphere. We show that under appropriate conditions this sequence has to terminate. In this case the Willmore surface either is the twistor projection of a holomorphic curve into CP 3 or the inversion of a minimal surface with planar ends in R 4 . These results give a unified explanation of previous work on the characterization of Willmore spheres and Willmore tori with non-trivial normal bundles by various authors.

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Section: The Willmore Sequencementioning

confidence: 99%

“…The second author thanks the Humboldt Foundation and the Matheon Center at TU-Berlin for their support. and projections [2]. This construction can be generalized to other target spaces such as Grassmannians [18] and Lie groups giving rise to the uniton transform of Uhlenbeck [17] .…”

Section: Introductionmentioning

confidence: 99%

Sequences of Willmore surfaces

Leschke

Pedit

2007

Math. Z.

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“…is uniquely determined by the induced metric, and <p can be constructed from its directrix <p Q : S 2 -> CP" by using arithmetical procedure [2].…”

Section: Preliminarymentioning

confidence: 99%

“…In [5] we studied equivariant minimal 3-spheres with constant (sectional) curvature c immersed in CP". Here the terminology 2 Zhen-Qi Li and An-Min Huang [2] 'equivariant' means that the immersion <p : S 3 ->• C P " is compatible with the Lie group structure on S 3 = SU (2), that is, there exists a homomorphism E : S 3 -> U(/i + 1) of Lie group such that (p = A o 7r 2 O E, where…”

Section: Preliminarymentioning

confidence: 99%

“…The bi-invariant metric ds 2 defines an inner product on su(2) in a natural way, and induces the inner product on su(2)* and su(2)* A 5u(2)* respectively. Without loss of generality we set a = -^/c and [dk x -(A, -ia) 2 coi -(|/x| [7] Constant curved minimal CR 3-spheres in CP" Comparing (3.3) with (2.14) and noting that co,, co 2 i, Po, Pi and p 2 are all real-valued 1-forms, we get Since <p is minimal, we have [5] (3.11) tr{Vco, -ipo ® co x + ipx ® co x -9~x 2 ® co} = 0, (3.12) tr{0 M ® to, + 0n ® a>} = 0 (A = 3 , . .…”

Section: Local Formulaementioning

confidence: 99%

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Constant curved minimal CR 3-spheres in CPⁿ

Zhen-qi¹,

Huang²

2005

J. Aust. Math. Soc.

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In this paper we prove that minimal 3-spheres of CR type with constant sectional curvature c in the complex projective space CP" are all equivariant and therefore the immersion is rigid. The curvature c of the sphere should be c = \/{m 2 -1) for some integer m > 2, and the full dimension is n = 2m 2 -3. An explicit analytic expression for such an immersion is given.2000 Mathematics subject classification: primary 53C42; secondary 53C55. Keywords and phrases: minimal, constant curvature, CR-submanifold, complex projective space. PreliminaryIn In this paper, we assume that N is the complex projective space CP" with constant holomorphic sectional curvature 4.The minimal surface theory in CP" has made a great progress over the past thirty years. For constant curved minimal 2-spheres in CP", the immersion

CP" by using arithmetical procedure [2].Up to now merely a few examples have been known for higher dimensional minimal submanifolds in CP". There are some examples of holomorphic submanifolds and Lagrangian minimal submanifolds [3,4,6]. In [5] we studied equivariant minimal 3-spheres with constant (sectional) curvature c immersed in CP". Here the terminology Project supported by the NSFC (10261006), the NSFJP (0211005) and the FANEDD (200217).

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