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 11
(a)
Mn
compac t 12
(b) M" noncompac t 15
3. Prescribing scalar curvature 16
4. Cauchy-Riemann manifold s 17
III.Ricci Curvature . 19
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
iii
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