Rigidity of pseudo-holomorphic curves of constant curvature in Grassmann manifolds

Authors:
Quo-Shin Chi and Yunbo Zheng

Journal:
Trans. Amer. Math. Soc. **313** (1989), 393-406

MSC:
Primary 53C42; Secondary 53C55

MathSciNet review:
992602

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Rigidity of minimal immersions of constant curvature in harmonic sequences generated by holomorphic curves in Grassmann manifolds is studied in this paper by lifting them to holomorphic curves in certain projective spaces. We prove that for such curves the curvature must be positive, and that all such simply connected curves in are generated by Veronese curves, thus generalizing Calabi's counterpart for holomorphic curves in . We also classify all holomorphic curves from the Riemann sphere into whose curvature is equal to into two families, which illustrates pseudo-holomorphic curves of positive constant curvature in are in general not unitarily equivalent, constracting to the fact that generic isometric complex submanifolds in a Kaehler manifold are congruent.

**[1]**John Bolton, Gary R. Jensen, Marco Rigoli, and Lyndon M. Woodward,*On conformal minimal immersions of 𝑆² into 𝐶𝑃ⁿ*, Math. Ann.**279**(1988), no. 4, 599–620. MR**926423**, 10.1007/BF01458531**[2]**Robert L. Bryant,*Minimal surfaces of constant curvature in 𝑆ⁿ*, Trans. Amer. Math. Soc.**290**(1985), no. 1, 259–271. MR**787964**, 10.1090/S0002-9947-1985-0787964-8**[3]**Eugenio Calabi,*Isometric imbedding of complex manifolds*, Ann. of Math. (2)**58**(1953), 1–23. MR**0057000****[4]**-,*Quelques applications de l'analyse aux surfaces d'aire minima*, Topics in Complex Manifolds, Presses de l'Université de Montréal, pp. 58-81.**[5]**Shiing Shen Chern,*On the minimal immersions of the two-sphere in a space of constant curvature*, Problems in analysis (Lectures at the Sympos. in honor of Salomon Bochner, Princeton Univ., Princeton, N.J., 1969) Princeton Univ. Press, Princeton, N.J., 1970, pp. 27–40. MR**0362151****[6]**Shiing Shen Chern and Jon G. Wolfson,*Harmonic maps of the two-sphere into a complex Grassmann manifold. II*, Ann. of Math. (2)**125**(1987), no. 2, 301–335. MR**881271**, 10.2307/1971312**[7]**J. Eells and J. C. Wood,*Harmonic maps from surfaces to complex projective spaces*, Adv. in Math.**49**(1983), no. 3, 217–263. MR**716372**, 10.1016/0001-8708(83)90062-2**[8]**Jost-Hinrich Eschenburg, Irwen Válle Guadalupe, and Renato de Azevedo Tribuzy,*The fundamental equations of minimal surfaces in 𝐶𝑃²*, Math. Ann.**270**(1985), no. 4, 571–598. MR**776173**, 10.1007/BF01455305**[9]**Mark L. Green,*Metric rigidity of holomorphic maps to Kähler manifolds*, J. Differential Geom.**13**(1978), no. 2, 279–286. MR**540947****[10]**P. Griffiths,*On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry*, Duke Math. J.**41**(1974), 775–814. MR**0410607****[11]**Phillip Griffiths and Joseph Harris,*Principles of algebraic geometry*, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR**507725****[12]**Gary R. Jensen,*Higher order contact of submanifolds of homogeneous spaces*, Lecture Notes in Mathematics, Vol. 610, Springer-Verlag, Berlin-New York, 1977. MR**0500648****[13]**Katsuei Kenmotsu,*On minimal immersions of 𝑅² into 𝑆^{𝑁}*, J. Math. Soc. Japan**28**(1976), no. 1, 182–191. MR**0405218****[14]**B. Lawson, Jr.,*Lectures on minimal submanifolds*, Vol. 1, Publish or Perish, Berkey, Calif., 1980.**[15]**Y. Zheng,*A quantization result of curvature for holomorphic curves in Grassmann manifolds*, preprint.**[16]**-,*Harmonic maps into Grassmann manifolds*, Thesis, Washington Univ., 1987.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
53C42,
53C55

Retrieve articles in all journals with MSC: 53C42, 53C55

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1989-0992602-9

Keywords:
Pseudo-holomorphic curves,
Grassmann manifolds,
Veronese curves

Article copyright:
© Copyright 1989
American Mathematical Society