Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/17074
Title: An initial value approach to rotationally symmetric harmonic maps
Authors: Cheung, LF
Law, CK
Keywords: Harmonic map
Hyperbolic space
Model space
Rotational symmetry
Issue Date: 2004
Publisher: Academic Press
Source: Journal of mathematical analysis and applications, 2004, v. 289, no. 1, p. 1-13 How to cite?
Journal: Journal of mathematical analysis and applications 
Abstract: We study the effect of the varying y′(0) on the existence and asymptotic behavior of solutions for the initial value problem {y″(r) + (n -1) f′(r)y′(r)/f(r) - (n - 1) g(y(r))g′(y(r))/f(r) 2 = 0, y(0) = 0, where f and g are some prescribed functions. Global solutions of this ODE on [0, ∞) represent rotationally symmetric harmonic maps, with possibly infinite energies, between certain class of Riemannian manifolds. By studying this ODE, we show among other things that (i) all rotationally symmetric harmonic maps from ℝ n to the hyperbolic space ℍ n blow up in a finite interval; (ii) all such harmonic maps from H n to R n are bounded; and (iii) a trichotomy phenomenon occurs for such harmonic maps from H n into itself, viz., they blow up in a finite interval, are the identity map, or are bounded according as the initial value y′(0) < 1, = 1, or > 1. Finally when n = 2, the above equation can be solved exactly by quadrature method. Our results supplement those of Ratio and Rigoli (J. Differential Equations 101 (1993) 15-27) and Tachikawa (Tokyo J. Math. 11 (1988) 311-316).
URI: http://hdl.handle.net/10397/17074
ISSN: 0022-247X
EISSN: 1096-0813
DOI: 10.1016/S0022-247X(03)00195-1
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