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Sökning: LAR1:gu > Chalmers tekniska högskola > Berman Robert 1976

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  • Berman, Robert, 1976- (författare)
  • A thermodynamical formalism for Monge-Ampere equations, Moser-Trudinger inequalities and Kahler-Einstein metrics
  • 2013
  • Ingår i: Advances in Mathematics. - 0001-8708. ; 248, s. 1254-1297
  • Tidskriftsartikel (refereegranskat)abstract
    • We develop a variational calculus for a certain free energy functional on the space of all probability measures on a Kahler manifold X. This functional can be seen as a generalization of Mabuchi's K-energy functional and its twisted versions to more singular situations. Applications to Monge-Ampere equations of mean field type, twisted Kahler-Einstein metrics and Moser-Trudinger type inequalities on Miller manifolds are given. Tian's alpha-invariant is generalized to singular measures, allowing in particular a proof of the existence of Kahler-Einstein metrics with positive Ricci curvature that are singular along a given anti-canonical divisor (which combined with very recent developments concerning Miller metrics with conical singularities confirms a recent conjecture of Donaldson). As another application we show that if the Calabi flow in the (anti-)canonical class exists for all times then it converges to a Kahler-Einstein metric, when a unique one exists, which is in line with a well-known conjecture. (C) 2013 Elsevier Inc. All rights reserved.
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  • Berman, Robert, 1976-, et al. (författare)
  • A variational approach to complex Monge-Ampere equations
  • 2013
  • Ingår i: Publications mathématiques. - 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.
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  • Berman, Robert, 1976- (författare)
  • Analytic torsion, vortices and positive Ricci curvature
  • 2010
  • Ingår i: preprint på arxiv.org.
  • Tidskriftsartikel (övrigt vetenskapligt)abstract
    • We characterize the global maximizers of a certain non-local functional defined on the space of all positively curved metrics on an ample line bundle L over a Kahler manifold X. This functional is an adjoint version, introduced by Berndtsson, of Donaldson's L-functional and generalizes the Ding-Tian functional whose critical points are Kahler-Einstein metrics of positive Ricci curvature. Applications to (1) analytic torsions on Fano manifolds (2) Chern-Simons-Higgs vortices on tori and (3) Kahler geometry are given. In particular, proofs of conjectures of (1) Gillet-Soul\'e and Fang (concerning the regularized determinant of Dolbeault Laplacians on the two-sphere) (2) Tarantello and (3) Aubin (concerning Moser-Trudinger type inequalities) in these three settings are obtained. New proofs of some results in Kahler geometry are also obtained, including a lower bound on Mabuchi's K-energy and the uniqueness result for Kahler-Einstein metrics on Fano manifolds of Bando-Mabuchi. This paper is a substantially extended version of the preprint arXiv:0905.4263 which it supersedes.
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  • Berman, Robert, 1976- (författare)
  • Bergman kernels and equilibrium measures for line bundles over projective manifolds
  • 2009
  • Ingår i: American Journal of Mathematics, Volume 131, Number 5, October 2009. ; s. 1485-1524
  • Tidskriftsartikel (refereegranskat)abstract
    • Let L be a holomorphic line bundle over a compact complex projective Hermitian manifold X. Any fixed smooth hermitian metric h on L induces a Hilbert space structure on the space of global holomorphic sections with values in the k th tensor power of L. In this paper various convergence results are obtained for the corresponding Bergman kernels (i.e. orthogonal projection kernels). The convergence is studied in the large k limit and is expressed in terms of the equilibrium metric h_e associated to h, as well as in terms of the Monge-Ampere measure of h on a certain support set. It is also shown that the equilibrium metric h_e is in the class C^{1,1} on the complement of the augmented base locus of L. For L ample these results give generalizations of well-known results concerning the case when the curvature of h is globally positive (then h_e=h). In general, the results can be seen as local metrized versions of Fujita's approximation theorem for the volume of L.
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  • Resultat 1-10 av 27
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