Composition of Gaussian Noises from Successive Convex Integrations

. In the context of isometric imbedding we consider the method of convex integration using Haar functions. Given a short map f 0 on [0 ; 1], under appropriate randomization we construct random isometric maps f n using convex integration. It is then shown that n 3 = 2 ( f n (cid:0) f 0 ) converges weakly to a Gaussian noise measure. We next consider the problem of composing the Gaussian noises from successive convex integrations since isometric imbedding for surfaces proceeds through similar steps. Some applications to approximate isometric imbeddings for two dimensional manifolds are also considered.