The segmentation of the early Drosophila embryo has proven to be a fertile ground for experimentalists and modellers. Until now, most models have assumed a one-dimensional geometry along a portion of the anterior-posterior axis, motivated by the nearly rotationally symmetrical observed patterns along this axis. While this approximation has been fruitful, new measurements on the whole surface of the blastoderm justify an extension of the models to the full geometry of the embryo. We develop a model for the gap gene network on the 2D surface of the blastoderm. Our model describes the dynamics of both mRNA and protein of 4 trunk gap genes expression during the cleavage cycles 12, 13 and 14A: hunchback, Kruppel, giant and knirps. The model takes as a regulatory inputs the protein expression of the maternal bicoid and caudal gradients, plus the zygotic tailless and huckebein. We formulate a reaction-diffusion model for the transcription regulatory network of gap genes on the curved surface of the embryo. Unknowns parameters of the model were estimated by non-linear optimization. We show that our model captures the main features of spatio-temporal patterning on the whole embryo. However, anterior domains, e.g. those in the giant gene, are difficult to reproduce, probably reecting oversimplifying assumptions or missing genes in the network. A detailed analysis of hunchback suggests that it has concentration dependent activity. We implement this effect and we show that it leads to a significant improvement of the best fit solution of our model. In conclusion solutions that capture both quantitative and qualitative aspects of the biological system were found. However theses solutions have some features that are not in agreement with the qualitative behavior of the biological system, indicative that we capture only a fraction of its properties