Regularity and Boundary Behavior of Solutions to Complex Monge–ampère Equations

1. Background 5 2. Plurisubharmonic functions 8 3. The complex Monge–Ampère operator 10 3.1. Bedford’s and Taylor’s definition of the complex Monge–Ampère operator 11 3.2. Cegrell’s definition of the complex Monge–Ampère operator 12 4. The Dirichlet problem for the complex Monge–Ampère operator 14 4.1. Boundary blow-up problems for the complex Monge–Ampère operator 17 4.2. Comparison principles 17 4.3. Regularity theory 18 4.4. Schauder theory 19 5. Applications of pluripotential theory to the theory of several complex variables 19 5.1. Boundary behavior of the Bergman kernel 20 5.2. Kähler–Einstein metrics 21 6. The results in the thesis 22 6.1. Paper I: Interior regularity of solutions to a complex Monge–Ampère equation 22 6.2. Paper II: The blow-up rate of solutions to boundary blow-up problems for the complex Monge–Ampère operator (joint work with J. Matero) 24 6.3. Paper III: Regularity and uniqueness of solutions to boundary blow-up problems for the complex Monge–Ampère operator 26 6.4. Paper IV: On the behavior of strictly plurisubharmonic functions near real hypersurfaces 28 6.5. Discussion 29 Acknowledgements 29 References 30