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ON ISOMETRIC MINIMAL IMMERSIONS FROM WARPED PRODUCTS INTO REAL SPACE FORMS

Published online by Cambridge University Press:  14 October 2002

Bang-Yen Chen
Bang-Yen Chen
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA (bychen@math.msu.edu)
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Abstract

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We establish a general sharp inequality for warped products in real space form. As applications, we show that if the warping function $f$ of a warped product $N_1\times_fN_2$ is a harmonic function, then

(1) every isometric minimal immersion of $N_1\times_fN_2$ into a Euclidean space is locally a warped-product immersion, and

(2) there are no isometric minimal immersions from $N_1\times_f N_2$ into hyperbolic spaces.

Moreover, we prove that if either $N_1$ is compact or the warping function $f$ is an eigenfunction of the Laplacian with positive eigenvalue, then $N_1\times_f N_2$ admits no isometric minimal immersion into a Euclidean space or a hyperbolic space for any codimension. We also provide examples to show that our results are sharp.

AMS 2000 Mathematics subject classification: Primary 53C40; 53C42; 53B25

Keywords

warped productswarped-product immersioninequalityminimal immersionsreal space formseigenfunction of the Laplacian
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 45 , Issue 3 , October 2002 , pp. 579 - 587
Copyright
Copyright © Edinburgh Mathematical Society 2002