EQUIVARIANT MINIMAL IMMERSIONS FROM S3 INTO CP3

doi:10.1007/s10473-019-0411-3

Abstract

Abstract: Associated with an immersion φ:S³ → CP³, we can define a canonical bundle endomorphism F:TS³→TS³ by the pull back of the Kähler form of CP³. In this article, related to F we study equivariant minimal immersions from S³ into CP³ under the additional condition (▽_XF)X=0 for all X ∈ ker (F). As main result, we give a complete classification of such kinds of immersions. Moreover, we also construct a typical example of equivariant non-minimal immersion φ:S³ → CP³ satisfying (▽_XF)X=0 for all X ∈ ker (F), which is neither Lagrangian nor of CR type.

Key words: Complex projective space, equivariant, minimal, 3-sphere

CLC Number:

53C24

Zejun HU, Jiabin YIN. EQUIVARIANT MINIMAL IMMERSIONS FROM S³ INTO CP³[J].Acta mathematica scientia,Series B, 2019, 39(4): 1065-1080.

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References

[1] Bejancu A. Gemetry of CR-submaniflds. Dordrecht:D Reidel Publishing Company, 1986
[2] Bolton J, Jensen G R, Rigoli M, Woodward L M. On conformal minimal immersions of S² into CPⁿ. Math Ann, 1988, 279:599-620
[3] Chen B Y, Dillen F, Verstraelen L, Vrancken L. An exotic totally real minimal immersions of S³ in CP³ and its characterization. Proc Roy Soc Edinb Sect A, 1996, 126:153-165
[4] Chen B Y, Dillen F, Verstraelen L, Vrancken L. Lagrangian isometric immersions of a real space form Mⁿ(c) into complex space form Mⁿ(c). Math Proc Camb Philos Soc, 1998, 124:107-125
[5] Chern S S, Wolfson J. Minimal surfaces by moving frames. Amer J Math, 1983, 105:59-83
[6] Dillen F, Li H Z, Vrancken L, Wang X F. Lagrangian submanifolds in complex projective space with parallel second fundamental form. Pac J Math, 2012, 255:79-115
[7] Fei J, Peng C K, Xu X W. Equivariant totally real 3-spheres in the complex projective space CPⁿ. Differ Geom Appl, 2012, 30:262-273
[8] Hu S, Li K. The minimal S³ with constant sectional curvature in CPⁿ. J Aust Math Soc, 2015, 99:63-75
[9] Hu Z J, Yin J B, Li Z Q. Equivariant CR minimal immersions from S³ into CPⁿ. Ann Global Anal Geom, 2018, 54:1-24
[10] Jenson G R, Liao R J. Families of flat minimal tori in CPⁿ. J Differ Geom, 1995, 42:113-132
[11] Li Z Q. Minimal S³ with constant curvature in CPⁿ, J Lond Math Soc, 2003, 68, 223-240
[12] Li Z Q, Huang A M. Constant curved minimal CR 3-spheres in CPⁿ. J Aust Math Soc, 2005, 79:1-10
[13] Li Z Q, Peng J W. Rigidity of 3-dimensional minimal CR-submanifolds with constant curvature in CPⁿ. Far East J Math Sci, 2009, 34:303-315
[14] Li Z Q, Tao Y Q. Equivariant Lagrangian minimal S³ in CP³. Acta Math Sin (Engl Ser), 2006, 22:1215-1220
[15] Li H Z, Vrancken L, Wang X F. A new characterization of the Berger sphere in complex projective space. J Geom Phys, 201592:129-139
[16] Mashimo K. Minimal immersions of 3-dimensional sphere into spheres. Osaka J Math, 1984, 21:721-732
[17] Naitoh H. Isotropic submanifolds with parallel second fundamental form in P^m(c). Osaka J Math, 1981, 18:427-464
[18] Naitoh H. Totally real parallel submanifolds in Pⁿ(c). Tokyo J Math, 19814:279-306
[19] Naitoh H. Parallel submanifolds of complex space forms. I. Nagoya Math J, 1983, 90:85-117; Ⅱ. ibid, 1983, 91:119-149
[20] Ogiue K. Differential geometry of Kähler submanifolds. Adv Math, 1974, 13:73-114

EQUIVARIANT MINIMAL IMMERSIONS FROM S³ INTO CP³

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