Pseudoholomorphic curves in S⁶ and S⁵

• Autores: Jost Hinrich Eschenburg, Theodoros Vlachos
• Localización: Revista de la Unión Matemática Argentina, ISSN 0041-6932, ISSN-e 1669-9637, Vol. 60, Nº. 2, 2019, págs. 517-537
• Idioma: inglés
• DOI: 10.33044/revuma.v60n2a16
• Enlaces
  • Texto completo (pdf)
• Resumen

  • The octonionic cross product on R7 induces a nearly K¨ahler structure on S 6 , the analogue of the K¨ahler structure of S 2 given by the usual (quaternionic) cross product on R3 . Pseudoholomorphic curves with respect to this structure are the analogue of meromorphic functions. They are (super-)conformal minimal immersions. We reprove a theorem of Hashimoto [Tokyo J. Math. 23 (2000), 137–159] giving an intrinsic characterization of pseudoholomorphic curves in S 6 and (beyond Hashimoto’s work) S 5 . Instead of the Maurer–Cartan equations we use an embedding theorem into homogeneous spaces (here: S 6 = G2/SU3) involving the canonical connection.

• Referencias bibliográficas
  • F. Belgun, A. Moroianu, Nearly K¨ahler 6-manifolds with reduced holonomy, Ann. Global Anal. Geom. 19 (2001), no. 4, 307–319. MR 1842572.
  • J. Bolton, L. Vrancken, L. Woodward, On almost complex curves in the nearly K¨ahler 6- sphere, Quart. J. Math. Oxford Ser. (2) 45 (1994),...
  • R. Bryant, Submanifolds and special structures on the octonians, J. Differential Geom. 17 (1982), no. 2, 185–232. MR 0664494.
  • E. Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Differential Geom. 1 (1967), 111–125. MR 0233294.
  • J.-H. Eschenburg, I.V. Guadalupe, R. Tribuzy, The fundamental equations of minimal surfaces in CP 2 , Math. Ann. 270 (1985), no. 4, 571–598....
  • J.-H. Eschenburg, R. Tribuzy, Existence and uniqueness of maps into affine homogeneous spaces, Rend. Sem. Mat. Univ. Padova 89 (1993), 11–18....
  • J.-H. Eschenburg, R. Tribuzy, Constant mean curvature surfaces in 4-space forms, Rend. Sem. Mat. Univ. Padova 79 (1988), 185–202. MR 0964030.
  • H. Hashimoto, J-holomorphic curves of a 6-dimensional sphere, Tokyo J. Math. 23 (2000), no. 1, 137–159. MR 1763509.
  • F. Lubbe, L. Sch¨afer, Pseudo-holomorphic curves in nearly K¨ahler manifolds, Differential Geom. Appl. 36 (2014), 24–43. MR 3262895.
  • G. Ricci, Lezioni sulla teoria delle superficie, Verona, Drucker, 1898, https://hdl.handle. net/2027/coo.31924059413561
  • Th. Vlachos, Congruence of minimal surfaces and higher fundamental forms, Manuscripta Math. 110 (2003), no. 1, 77–91. MR 1951801.
  • Th. Vlachos, Minimal surfaces, Hopf differentials and the Ricci condition, Manuscripta Math. 126 (2008), no. 2, 201–230. MR 2403186.
  • Th. Vlachos, Exceptional minimal surfaces in spheres, Manuscripta Math. 150 (2016), no. 1-2, 73–98. MR 3483170.