Mirror Symmetry, Picard-Fuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to establish mirror symmetry of Calabi-Yau spaces for which the mirror manifold had been unavailable in previous constructions. Mirror maps and Yukawa couplings are explicitly given for several examples with two and three moduli.

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Introduction. The classical Schwarz-Pick lemma states that any holomorphic map of the unit disk into itself decreases the Poincare metric. Later Ahlfors generalized this lemma to holomorphic mappings between two Riemann surfaces where curvatures of these Riemann surfaces were used in a very explicit way. More recently, Chern initiated the study of holomorphic mappings between higher-dimensional complex manifold by generalizing the Ahlfors lemma to these spaces. Then this lemma was further extended by Kobayashi, Griffiths, Wu and others. It plays a very important role in their theory. In this note, we shall prove the following generalization of the Schwarz lemma.

This work is intended as both an introductory text and a reference book for those interested in studying several complex variables in the context of partial differential equations. Significant progress has been made in the study of Cauchy-Riemann and tangential Cauchy-Riemann operators; this progress greatly influenced the development of PDEs and several complex variables.

Let M be a complete two-dimensional surface immersed into the three-dimensional Euclidean space. Then a classical theorem of Hilbert says that when the curvature of M is a non-zero constant, M must be the sphere. On the other hand, when the curvature of M is zero, a theorem of Hartman-Nirenberg [4] says that M must be a plane or a cylinder. These two theorems complete the classification of complete surfaces with constant curvature in R 3.