Contents Preface v Notation vi i I. Gaussia n Curvature 1 1. Surfaces i n R3 1 2. Prescribing the curvature form o n a surface 3 3. Prescribin g the Gaussian curvature on a surface 3 (a) Compact surface s 4 (b) Noncompact surface s 6 II. Scala r Curvature 9 1. Topological obstructions . . 9 2. Pointwise conformal deformation s an d the Yamabe problem 1 1 (a) Mn compac t 1 2 (b) M" noncompac t 1 5 3. Prescribing scalar curvature 1 6 4. Cauchy-Riemann manifold s 1 7 III.Ricci Curvature . 1 9 1. Local solvability of Ric(g ) = R ij 2 0 2. Local smoothness of metrics 2 1 3. Global topologica l obstructions 2 1 4. Uniqueness, nonexistence 2 3 5. Einstein metrics on 3-manifolds 2 3 6. Kahler manifolds 2 6 (a) Kahler geometry 2 6 (b) Calabi's problem and Kahler-Einstein metric s 2 7 (c) Another variationa l problem 3 1 IV. Boundary Value Problems 3 3 1. Surfaces with constant mean curvature and Rellich' s problem 3 3 2. Some other boundary value problems 3 6 (a) Graphs with prescribed mean curvature 3 6 (b) Graphs with prescribed Gauss curvature 3 8 3. The C 2 + a estimat e at the boundary 3 9 Some Open Problems 4 7 References 5 1 iii

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