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| 12. |
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| 13. |
- Berman, Robert, 1976
(författare)
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On the optimal regularity of weak geodesics in the space of metrics on a polarized manifold
- 2017
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Ingår i: Analysis Meets Geometry. (Trends in Mathematics). Andersson M., Boman J., Kiselman C., Kurasov P., Sigurdsson R. (eds). - Cham : Springer. - 2297-0215 .- 2297-024X. - 9783319524719 ; , s. 111-120
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Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- © 2017 Springer International Publishing. Let (X,L) be a polarized compact manifold, i.e., L is an ample line bundle over X and denote by ℋ the infinite-dimensional space of all positively curved Hermitian metrics on L equipped with the Mabuchi metric. In this short note we show, using Bedford–Taylor type envelope techniques developed in the authors previous work [3], that Chen’s weak geodesic connecting any two elements in ℋ are C 1,1 -smooth, i.e., the real Hessian is bounded, for any fixed time t, thus improving the original bound on the Laplacians due to Chen. This also gives a partial generalization of Blocki’s refinement of Chen’s regularity result. More generally, a regularity result for complex Monge–Ampère equations over X × D, for D a pseudoconvex domain in ℂ n is given.
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| 14. |
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| 15. |
- Berman, Robert, 1976
(författare)
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Relative Kahler-Ricci flows and their quantization
- 2010
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Ingår i: preprint på arxiv.org.
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- Let X be a complex manifold fibered over the base S and let L be a relatively ample line bundle over X. We define relative Kahler-Ricci flows on the space of all Hermitian metrics on L with relatively positive curvature. Mainly three different settings are investigated: the case when the fibers are Calabi-Yau manifolds and the case when L is the relative (anti-) canonical line bundle. The main theme studied is whether posivity in families is preserved under the flows and its relation to the variation of the moduli of the complex structures of the fibres. The quantization of this setting is also studied, where the role of the Kahler-Ricci flow is played by Donaldson's iteration on the space of all Hermitian metrics on the finite rank vector bundle over S defined as the zeroth direct image induced by the fibration. Applications to the construction of canonical metrics on relative canonical bumdles and Weil-Petersson geometry are given. Some of the main results are a parabolic analogue of a recent elliptic equation of Schumacher and the convergence towards the K\"ahler-Ricci flow of Donaldson's iteration in a certain double scaling limit.
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| 16. |
- Berman, Robert, 1976
(författare)
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Sharp asymptotics for Toeplitz determinants, fluctuations and the gaussian free field on a Riemann surface
- 2011
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Ingår i: preprint på arxiv.org.
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- We consider canonical determinantal random point processes with N particles on a compact Riemann surface X defined with respect to the constant curvature metric. In the higher genus (hyperbolic) cases these point processes may be defined in terms of automorphic forms. We establish strong exponential concentration of measure type properties involving Dirichlet norms of linear statistics. This gives an optimal Central Limit Theorem (CLT), saying that the fluctuations of the corresponding empirical measures converge, in the large N limt, towards the Laplacian of the Gaussian free field on X in the strongest possible sense. The CLT is also shown to be equivalent to a new sharp strong Szeg\"o type theorem for Toeplitz determinants in this context. One of the ingredients in the proofs are new Bergman kernel asymptotics providing exponentially small error terms in a constant curvature setting.
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