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