The purpose of this chapter is to provide the necessary background material about pseudoholomorphic curves. We cover punctured holomorphic curves with or without boundary in the symplectization of a contact manifold as in H. Hofer’s 1993 article and in papers by the author. The behavior near a puncture (boundary or interior) is discussed in detail. We also cover Gromov’s Isoperimetric inequality, Monotonicity Lemma and the theorem about removal of singularities. Generalizations for curves with punctures and curves with boundary are explained as well. The chapter ends with a discussion of how pseudoholomorphic curves can degenerate and form holomorphic buildings. A few ‘folk’ results well known to specialists in the area but without proofs in the literature are also provided in this chapter.
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