International Journal of Mathematics and Mathematical Sciences
Volume 30 (2002), Issue 12, Pages 709-715
doi:10.1155/S0161171202107058
Harmonicity of horizontally conformal maps and spectrum of the Laplacian
Department of Mathematics, Myong Ji University, San 38-2, Namdong, Yongin, Kyunggi 449-728, Korea
Received 18 July 2001; Revised 29 December 2001
Copyright © 2002 Gabjin Yun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We discuss the harmonicity of horizontally conformal maps and
their relations with the spectrum of the Laplacian. We prove that if Φ:M→N is a horizontally conformal map such that the tension field is divergence free, then Φ is harmonic. Furthermore, if N is noncompact, then Φ must be constant. Also we show that the projection of a warped product manifold onto the first component is harmonic if and only if the warping function is constant. Finally, we describe a characterization for a horizontally conformal map with a constant dilation preserving an eigenfunction.