Mathematics – Complex Variables
Scientific paper
2006-12-21
Mathematics
Complex Variables
Final version (34 pages). To appear in Journal of Functional Analysis
Scientific paper
We study degenerate complex Monge-Amp\`ere equations on a compact K\"ahler manifold $(X,\omega)$. We show that the complex Monge-Amp\`ere operator $(\omega + dd^c \cdot)^n$ is well-defined on the class ${\mathcal E}(X,\omega)$ of $\omega$-plurisubharmonic functions with finite weighted Monge-Amp\`ere energy. The class ${\mathcal E}(X,\omega)$ is the largest class of $\omega$-psh functions on which the Monge-Amp\`ere operator is well-defined and the comparison principle is valid. It contains several functions whose gradient is not square integrable. We give a complete description of the range of the Monge-Amp\`ere operator $(\omega +dd^c \cdot)^n$ on ${\mathcal E}(X,\omega)$, as well as on some of its subclasses. We also study uniqueness properties, extending Calabi's result to this unbounded and degenerate situation, and we give applications to complex dynamics and to the existence of singular K\"ahler-Einstein metrics.
Guedj Vincent
Zeriahi Ahmed
No associations
LandOfFree
The weigthed Monge-Ampère energy of quasiplurisubharmonic functions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The weigthed Monge-Ampère energy of quasiplurisubharmonic functions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The weigthed Monge-Ampère energy of quasiplurisubharmonic functions will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-22522