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

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1.
  • 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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6.
  • Berman, Robert, 1976-, et al. (författare)
  • Bergman Geodesics
  • 2012
  • Ingår i: Lecture notes in mathematics. - 0075-8434. - 978-3-642-23669-3978-3-642-23668-6 ; 2038, s. 283-302
  • Tidskriftsartikel (refereegranskat)
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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. - 0002-9327. ; 131:5, 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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  • Berman, Robert, 1976- (författare)
  • Bergman kernels and equilibrium measures for polarized pseudoconcave domains
  • 2010
  • Ingår i: International Journal of Mathematics. - 0129-167X. ; 21:1, s. 77
  • Tidskriftsartikel (refereegranskat)abstract
    • Let X be a domain in a closed polarized complex manifold (Y, L), where L is a (semi-) positive line bundle over Y. Any given Hermitian metric on L induces by restriction to X a Hilbert space structure on the space of global holomorphic sections on Y with values in the k-th tensor power of L (also using a volume form omega(n) on X). In this paper the leading large k asymptotics for the corresponding Bergman kernels and metrics are obtained in the case when X is a pseudo-concave domain with smooth boundary (under a certain compatibility assumption). The asymptotics are expressed in terms of the curvature of L and the boundary of X. The convergence of the Bergman metrics is obtained in a more general setting where (X, omega(n)) is replaced by any measure satisfying a Bernstein-Markov property. As an application the (generalized) equilibrium measure of the polarized pseudo-concave domain X is computed explicitly. Applications to the zero and mass distribution of random holomorphic sections and the eigenvalue distribution of Toeplitz operators will be described elsewhere.
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