Geometriae Dedicata 65: 47–57, 1997. 47 c 1997 Kluwer Academic Publishers. Printed in the Netherlands. Classification of Isometric Immersions of the Hyperbolic Space 2 into 3 HU ZE-JUN 1 and ZHAO GUO-SONG 2 1 Department of Mathematics, Zhengzhou University, Zhengzhou, Henan 450052, P.R. China 2 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P.R. China (Received: 7 July 1995) Abstract. We transform the problem of determining isometric immersions from 2 1 into 3 1 into that of solving an elliptic Monge–Amp` ere equation on the unit disc. Then we classify isometric immersions which possess bounded principal curvatures. Mathematics Subject Classifications (1991): 53C21, 53C42, 53C45. Key words: Hyperbolic space, isometric immersion, principal curvature, Monge–Amp` ere equation. 1. Introduction Let 0 be a hyperbolic space with constant sectional curvature . Let 1 1 be a Minkowski space, i.e. consider 1 with coordinates 1 1 and equipped with a Minkowski inner product such that 1 1 1 for any 1 1 1 1 in 1 . The Cayley model of consists of the hypersurface : 1 1 0 with the metric induced from 1 1 . Let us consider an isometric immersion : 2 3 . Without loss of generality we assume that 1. Take the hyperplane : 3 1 in 3 1 and consider the central projection of : 2 1 2 2 2 3 1 3 0 from the origin of 3 into , then is mapped in a one-to-one fashion onto an open unit ball : 2 1 2 2 1. The map is given by : 1 2 3 1 2 where 3 1 2 1 2 2 3 12 This research has been supported by the National Natural Science Foundation of China and the Tianyuan Mathematics Foundation of China.