On Isometric Minimal Immersions From Warped Products Into Real Space Forms

Abstract 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 N₁fN₂ is a harmonic function, then (1) every isometric minimal immersion of N₁fN₂ into a Euclidean space is locally a warped-product immersion, and (2) there are no isometric minimal immersions from N₁f N₂ into hyperbolic spaces. Moreover, we prove that if either N₁ is compact or the warping function f is an eigenfunction of the Laplacian with positive eigenvalue, then N₁f N₂ 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