Math 8230, Spring 2010 Lecture Notes

1. Introductory remarks 1 2. Energy and area 3 3. Exact Lagrangian submanifolds and the non-squeezing theorem 7 4. Sobolev spaces 9 5. The linear Cauchy–Riemann operator 20 5.1. Proof of the Calderón–Zygmund Theorem for p , 2 26 6. Local properties of J-holomorphic curves 30 6.1. Smoothness 31 6.2. The Carleman Similarity Principle and its applications 35 6.3. Isolation of intersection points and somewhere injectivity 40 7. Manifold structures on spaces of J-holomorphic curves 44 7.1. Banach manifolds 44 7.2. The linearization of the Cauchy–Riemann equation 46 7.3. Generic surjectivity 50 7.4. Variations on Theorem 7.1 53 8. Compactness 55 8.1. The isoperimetric inequality and the monotonicity formula 56 8.2. Removal of singularities and Gromov’s Schwarz Lemma 60 8.3. Bubbling 63 9. Nonsqueezing revisited 70 References 73