# 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

## 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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