CONFLUENCE OF SINGULAR POINTS OF ORDINARY DIFFERENTIAL EQUATIONS OF FUCHSIAN TYPE INDUCED BY DEFORMATION OF TWO-DIMENSIONAL HYPERBOLIC CONE-MANIFOLD STRUCTURES CONFLUENCE OF SINGULAR POINTS OF ORDINARY DIFFERENTIAL EQUATIONS OF FUCHSIAN TYPE INDUCED BY DEFORMATION OF TWO-DIMENSIONAL HYPERBOLIC CONE-MANIFOLD STRUCTURES

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Author(s)

    • FUJII Michihiko
    • Department of Mathematics Graduate School of Science Kyoto University

Abstract

Let $ \{ \sigma_t \}_t \in (-\infty, \infty) $ be a one-parameter family of hyperbolic Riemannian metrics on an open annulus which is continuouswith respect to the Gromov-Hausdorff topology. We consider a system $ E_t $ of ordinary differential equations with singular points which depends on the Riemannian metric $ \sigma_t $. If $ t \neq 0 $, all of the singular points of $ E_t $ are regular. If $ t = 0 $, $ E_0 $ has an irregular singular point. In this paper, we investigate the behavior of the singular points of $ E_t $. We show that a regular singular point of $ E_t $, together with another regular singular point of $ E_t $, becomes the irregular singular point of $ E_0 $ as $ t $ $ (>0) $ tends to zero and that the irregular singular point of $ E_0 $ becomes a non-singular point of $ E_t $ as $ t $ decreases from zero.

Let {σ<SUB><I>t</I></SUB>}<SUB><I>t</I>∈(-∞,∞)</SUB> be a one-parameter family of hyperbolic Riemannian metrics on an open annulus which is continuouswith respect to the Gromov-Hausdorff topology. We consider a system <I>E</I><SUB><I>t</I></SUB> of ordinary differential equations with singular points which depends on the Riemannian metric σ<SUB><I>t</I></SUB>. If <I>t</I> ≠ 0, all of the singular points of <I>E</I><SUB><I>t</I></SUB> are regular. If <I>t</I> = 0, <I>E</I><SUB>0</SUB> has an irregular singular point. In this paper, we investigate the behavior of the singular points of <I>E</I><SUB><I>t</I></SUB> . We show that a regular singular point of <I>E</I><SUB><I>t</I></SUB> , together with another regular singular point of <I>E</I><SUB><I>t</I></SUB> , becomes the irregular singular point of <I>E</I><SUB>0</SUB> as <I>t</I> (›0) tends to zero and that the irregular singular point of <I>E</I><SUB>0</SUB> becomes a non-singular point of <I>E</I><SUB><I>t</I></SUB> as <I>t</I> decreases from zero.

Journal

  • Kyushu Journal of Mathematics

    Kyushu Journal of Mathematics 61(1), 21-34, 2007

    Faculty of Mathematics, Kyushu University

Codes

  • NII Article ID (NAID)
    130000063182
  • NII NACSIS-CAT ID (NCID)
    AA10994346
  • Text Lang
    ENG
  • Article Type
    Departmental Bulletin Paper
  • ISSN
    1340-6116
  • Data Source
    IR  J-STAGE 
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