Skip to main content
×
Home
    • Aa
    • Aa

Holomorphic Legendrian curves

  • Antonio Alarcón (a1), Franc Forstnerič (a2) and Francisco J. López (a3)
Abstract

In this paper we study holomorphic Legendrian curves in the standard holomorphic contact structure on $\mathbb{C}^{2n+1}$ for any $n\in \mathbb{N}$ . We provide several approximation and desingularization results which enable us to prove general existence theorems, settling some of the open problems in the subject. In particular, we show that every open Riemann surface $M$ admits a proper holomorphic Legendrian embedding $M{\hookrightarrow}\mathbb{C}^{2n+1}$ , and we prove that for every compact bordered Riemann surface $M={M\unicode[STIX]{x0030A}}\,\cup \,bM$ there exists a topological embedding $M{\hookrightarrow}\mathbb{C}^{2n+1}$ whose restriction to the interior is a complete holomorphic Legendrian embedding ${M\unicode[STIX]{x0030A}}{\hookrightarrow}\mathbb{C}^{2n+1}$ . As a consequence, we infer that every complex contact manifold $W$ carries relatively compact holomorphic Legendrian curves, normalized by any given bordered Riemann surface, which are complete with respect to any Riemannian metric on $W$ .

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

R. Abraham , Transversality in manifolds of mappings , Bull. Amer. Math. Soc. 69 (1963), 470474; MR 0149495.

A. Alarcón , B. Drinovec Drnovšek , F. Forstnerič and F. J. López , Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves , Proc. Lond. Math. Soc. (3) 111 (2015), 851886; MR 3407187.

A. Alarcón and F. Forstnerič , Every bordered Riemann surface is a complete proper curve in a ball , Math. Ann. 357 (2013), 10491070; MR 3118624.

A. Alarcón and F. Forstnerič , Null curves and directed immersions of open Riemann surfaces , Invent. Math. 196 (2014), 733771; MR 3211044.

A. Alarcón and F. Forstnerič , The Calabi–Yau problem, null curves, and Bryant surfaces , Math. Ann. 363 (2015), 913951; MR 3412347.

A. Alarcón , F. Forstnerič and F. J. López , Embedded minimal surfaces in ℝ n , Math. Z. 283 (2016), 124; MR 3489056.

A. Alarcón and F. J. López , Minimal surfaces in ℝ3 properly projecting into ℝ2 , J. Differential Geom. 90 (2012), 351381; MR 2916039.

A. Alarcón and F. J. López , Null curves in ℂ3 and Calabi–Yau conjectures , Math. Ann. 355 (2013), 429455; MR 3010135.

A. Alarcón and F. J. López , Properness of associated minimal surfaces , Trans. Amer. Math. Soc. 366 (2014), 51395154; MR 3240920.

A. Alarcón and F. J. López , Approximation theory for nonorientable minimal surfaces and applications , Geom. Topol. 19 (2015), 10151062; MR 3336277.

A. Alarcón and F. J. López , Complete bounded embedded complex curves in ℂ2 , J. Eur. Math. Soc. (JEMS) 18 (2016), 16751705; MR 3519537.

R. L. Bryant , Conformal and minimal immersions of compact surfaces into the 4-sphere , J. Differential Geom. 17 (1982), 455473; MR 679067.

B. Drinovec Drnovšek and F. Forstnerič , The Poletsky–Rosay theorem on singular complex spaces , Indiana Univ. Math. J. 61 (2012), 14071423; MR 3085613.

Y. Eliashberg , Classification of overtwisted contact structures on 3-manifolds , Invent. Math. 98 (1989), 623637; MR 1022310.

F. Forstnerič , Stein manifolds and holomorphic mappings: the homotopy principle in complex analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, A Series of Modern Surveys in Mathematics, vol. 56 (Springer, Heidelberg, 2011); MR 2975791.

F. Forstnerič and E. F. Wold , Bordered Riemann surfaces in ℂ2 , J. Math. Pures Appl. (9) 91 (2009), 100114; MR 2487902.

H. Geiges , An introduction to contact topology, Cambridge Studies in Advanced Mathematics, vol. 109 (Cambridge University Press, Cambridge, 2008); MR 2397738.

H. Geiges , Contact structures and geometric topology , in Global differential geometry, Springer Proceedings in Mathematics, vol. 17 (Springer, Heidelberg, 2012), 463489; MR 3289851.

M. Gromov , Carnot–Carathéodory spaces seen from within , in Sub-Riemannian geometry, Progress in Mathematics, vol. 144 (Birkhäuser, Basel, 1996), 79323; MR 1421823.

R. C. Gunning and R. Narasimhan , Immersion of open Riemann surfaces , Math. Ann. 174 (1967), 103108; MR 0223560 (36 #6608).

C. LeBrun , Fano manifolds, contact structures, and quaternionic geometry , Internat. J. Math. 6 (1995), 419437; MR 1327157.

J. M. Landsberg and L. Manivel , Legendrian varieties , Asian J. Math. 11 (2007), 341359; MR 2372722.

F. Martín , M. Umehara and K. Yamada , Flat surfaces in hyperbolic 3-space whose hyperbolic Gauss maps are bounded , Rev. Mat. Iberoam. 30 (2014), 309316; MR 3186941.

J. Moser , On the volume elements on a manifold , Trans. Amer. Math. Soc. 120 (1965), 286294; MR 0182927.

H. J. Sussmann , Orbits of families of vector fields and integrability of distributions , Trans. Amer. Math. Soc. 180 (1973), 171188; MR 0321133.

H. J. Sussmann , Orbits of families of vector fields and integrability of systems with singularities , Bull. Amer. Math. Soc. 79 (1973), 197199; MR 0310922.

P. Yang , Curvatures of complex submanifolds of C n , J. Differential Geom. 12 (1978), 499511; MR 512921.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 5
Total number of PDF views: 14 *
Loading metrics...

Abstract views

Total abstract views: 78 *
Loading metrics...

* Views captured on Cambridge Core between 29th June 2017 - 19th September 2017. This data will be updated every 24 hours.