| 1. |
- Berman, Robert, 1976, et al.
(författare)
-
A variational approach to complex Monge-Ampere equations
- 2013
-
Ingår i: Publications mathématiques. - : Springer Science and Business Media LLC. - 0073-8301. ; 117:1, s. 179-245
-
Tidskriftsartikel (refereegranskat)abstract
- We show that degenerate complex Monge-Ampère equations in a big cohomology class of a compact Kähler manifold can be solved using a variational method, without relying on Yau’s theorem. Our formulation yields in particular a natural pluricomplex analogue of the classical logarithmic energy of a measure. We also investigate Kähler-Einstein equations on Fano manifolds. Using continuous geodesics in the closure of the space of Kähler metrics and Berndtsson’s positivity of direct images, we extend Ding-Tian’s variational characterization and Bando-Mabuchi’s uniqueness result to singular Kähler-Einstein metrics. Finally, using our variational characterization we prove the existence, uniqueness and convergence as k→∞ of k-balanced metrics in the sense of Donaldson both in the (anti)canonical case and with respect to a measure of finite pluricomplex energy.
|
|
| 2. |
- Berman, Robert, 1976, et al.
(författare)
-
Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties
- 2019
-
Ingår i: Journal für die Reine und Angewandte Mathematik. - : Walter de Gruyter GmbH. - 1435-5345 .- 0075-4102. ; 2019:751, s. 27-89
-
Tidskriftsartikel (refereegranskat)abstract
- We prove the existence and uniqueness of Kähler-Einstein metrics on Q-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on the convergence of the normalized Kähler-Ricci flow, and of Keller, Rubinstein on its discrete version, Ricci iteration. In the special case of (non-singular) Fano manifolds, our results on Ricci iteration yield smooth convergence without any additional condition, improving on previous results. Our result for the Kähler-Ricci flow provides weak convergence independently of Perelman's celebrated estimates.
|
|
| 3. |
|
|
| 4. |
|
|
| 5. |
|
|
| 6. |
- Åhag, Per, et al.
(författare)
-
Partial pluricomplex energy and integrability exponents of plurisubharmonic functions
- 2009
-
Ingår i: Advances in Mathematics. - 0001-8708 .- 1090-2082. ; 222:6, s. 2036-2058
-
Tidskriftsartikel (refereegranskat)abstract
- We first prove a quantitative estimate of the volume of the sublevel sets of a plurisubharmonic function in a hyperconvex domain with boundary values 0 (in a quite general sense) in terms of its Monge–Ampère mass in the domain. Then we deduce a sharp sufficient condition on the Monge–Ampère mass of such a plurisubharmonic function φ for exp(−2φ) to be globally integrable as well as locally integrable.
|
|
