A Geometric Application of Runge's Theorem

• Ildefonso Castro-Infantes ^[1]
  1. [1] Universidad de Granada

     Universidad de Granada

     Granada, España

     
• Localización: Reports@SCM: an electronic journal of the Societat Catalana de Matemàtiques, ISSN-e 2385-4227, Vol. 2, Nº. 1, 2016, págs. 21-32
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen
  • català

    En aquest article donem una demostració simple de l'existència d'aplicacions harmòniques de qualsevol superfície de Riemann cap al pla complex C ≡ R2. La nostra eina principal és la teoria d'aproximació per funcions holomorfes en superfícies de Riemann.

  • English

    In this article we give a simple proof of the existence of proper harmonic maps from any open Riemann surface into the complex plane C ≡ R2. Our main tool will be the Approximation Theory by holomorphic functions on Riemann surfaces.

• Referencias bibliográficas
  • A. Alarcon and J.A. Galvez, “Proper harmonicm aps from hyperbolic Riemann surfaces into the euclidean plane”, Results Math.60(1–4)(2011),...
  • A. Alarcon and F.J. Lopez, “Minimal surfaces inR3properly projecting into R2”,J. DifferentialGeom.90(3) (2012), 351–381.
  • A. Alarcon and F.J. Lopez, “Properness of associated minimal surfaces”,Trans. Amer. Math.Soc.366(10) (2014), 5139–5154.
  • R.B. Andrist and E.F. Wold, “Riemann surfaces in stein manifolds with density property”,preprint. arXiv:1106.4416v1.
  • E. Bishop, “Subalgebras of functions on a Riemann surface”,Pacific J. Math.8(1958), 29–50.
  • F. Forstneric and J. Globevnik, “Proper holo-morphic disks in C2”,Math. Res. Lett.8(3)(2001), 257–274.
  • A. Grigoryan, “Analytic and geometric background of recurrence and non-explosion of theBrownian motion on Riemannian manifolds”,Bull....
  • E. Heinz, “Uber die Losungen der Mini-malflachengleichung” (german),Nachr. Akad.Wiss. G ̈ottingen. Math.-Phys. Kl. Math.-Phys.-Chem....
  • S.N. Mergelyan, “On the representation of functions by series of polynomials on closed sets”,Doklady Akad. Nauk SSSR78(1951),405–408.
  • C. Runge, “Zur Theorie der Analytischen Functionen” (german),Acta Math.6(1885), 245–248.
  • R. Schoen and S.T. Yau, “Lectures on harmonic maps”,Conference Proceedings and Lecture Notes in Geometry and Topology, II. Inter-national...
  • H.M. Farkas and I. Kra, “Riemann surfaces”(second edition),Graduate Texts in Mathemat-ics71. Springer-Verlag, New York, 1992.