On Hamiltonian minimal submanifolds in the space of oriented geodesics in real space forms

Nikos Georgiou, Guillermo A. Lobos

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1 Citation (Scopus)

Abstract

We prove that a deformation of a hypersurface in an (n + 1)-dimensional real space form (Formula presented.) induces a Hamiltonian variation of the normal congruence in the space (Formula presented.) of oriented geodesics. As an application, we show that every Hamiltonian minimal submanifold in (Formula presented.) (resp. (Formula presented.) with respect to the (para-)Kähler Einstein structure is locally the normal congruence of a hypersurface (Formula presented.) in (Formula presented.) (resp. (Formula presented.) that is a critical point of the functional (Formula presented.) , where ki denote the principal curvatures of (Formula presented.) and (Formula presented.). In addition, for (Formula presented.) , we prove that every Hamiltonian minimal surface in (Formula presented.) (resp. (Formula presented.), with respect to the (para-)Kähler conformally flat structure, is the normal congruence of a surface in (Formula presented.) (resp. (Formula presented.) that is a critical point of the functional (Formula presented.) (resp. (Formula presented.), where H and K denote, respectively, the mean and Gaussian curvature of (Formula presented.).

Original languageEnglish
Pages (from-to)285-293
Number of pages9
JournalArchiv der Mathematik
Volume106
Issue number3
DOIs
Publication statusPublished - 01 Mar 2016

Keywords

  • Hamiltonian minimal submanifolds
  • Kähler structures
  • Real space forms
  • Space of oriented geodesics

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