| 1. |
- Berman, Robert, 1976, et al.
(författare)
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Convexity of the extended K-energy and the large time behavior of the weak Calabi flow
- 2017
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Ingår i: Geometry and Topology. - : Mathematical Sciences Publishers. - 1465-3060 .- 1364-0380. ; 21:5, s. 2945-2988
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Tidskriftsartikel (refereegranskat)abstract
- © 2017, Mathematical Sciences Publishers. All rights reserved. Let (X, ω) be a compact connected Kähler manifold and denote by(ε p , d p ) the metric completion of the space of Kähler potentials H ω with respect to the L p -type path length metric d p . First, we show that the natural analytic extension of the (twisted) Mabuchi K-energy to ε p is a d p -1sc functional that is convex along finite-energy geodesics. Second, following the program of J Streets, we use this to study the asymptotics of the weak (twisted) Calabi flow inside the CAT(0) metric space (ε 2 , d 2 ). This flow exists for all times and coincides with the usual smooth (twisted) Calabi flow whenever the latter exists. We show that the weak (twisted) Calabi flow either diverges with respect to the d 2 -metric or it d 1 -converges to some minimizer of the K-energy inside ε 2 . This gives the first concrete result about the long-time convergence of this flow on general Kähler manifolds, partially confirming a conjecture of Donaldson. We investigate the possibility of constructing destabilizing geodesic rays asymptotic to diverging weak (twisted) Calabi trajectories, and give a result in the case when the twisting form is Kähler. Finally, when a cscK metric exists in H ω , our results imply that the weak Calabi flow d 1 -converges to such a metric.
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| 2. |
- Berman, Robert, 1976, et al.
(författare)
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Convexity of the K-energy on the space of Kähler metrics and uniqueness of extremal metrics
- 2017
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Ingår i: Journal of the American Mathematical Society. - : American Mathematical Society (AMS). - 0894-0347 .- 1088-6834. ; 30:4, s. 1165-1196
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Tidskriftsartikel (refereegranskat)abstract
- We establish the convexity of Mabuchi's K-energy functional along weak geodesics in the space of Kähler potentials on a compact Kähler manifold, thus confirming a conjecture of Chen, and give some applications in Kähler geometry, including a proof of the uniqueness of constant scalar curvature metrics (or more generally extremal metrics) modulo automorphisms. The key ingredient is a new local positivity property of weak solutions to the homogeneous Monge-Ampère equation on a product domain, whose proof uses plurisubharmonic variation of Bergman kernels.
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| 3. |
- Berman, Robert, 1976
(författare)
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From Monge–Ampère equations to envelopes and geodesic rays in the zero temperature limit
- 2019
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Ingår i: Mathematische Zeitschrift. - : Springer Science and Business Media LLC. - 1432-1823 .- 0025-5874. ; 291:1-2, s. 365-394
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Tidskriftsartikel (refereegranskat)abstract
- Let (X, θ) be a compact complex manifold X equipped with a smooth (but not necessarily positive) closed (1, 1)-form θ. By a well-known envelope construction this data determines, in the case when the cohomology class [θ] is pseudoeffective, a canonical θ-psh function u θ . When the class [θ] is Kähler we introduce a family u β of regularizations of u θ , parametrized by a large positive number β, where u β is defined as the unique smooth solution of a complex Monge–Ampère equation of Aubin–Yau type. It is shown that, as β→ ∞, the functions u β converge to the envelope u θ uniformly on X in the Hölder space C 1,α (X) for any α∈] 0 , 1 [(which is optimal in terms of Hölder exponents). A generalization of this result to the case of a nef and big cohomology class is also obtained and a weaker version of the result is obtained for big cohomology classes. The proofs of the convergence results do not assume any a priori regularity of u θ . Applications to the regularization of ω-psh functions and geodesic rays in the closure of the space of Kähler metrics are given. As briefly explained there is a statistical mechanical motivation for this regularization procedure, where β appears as the inverse temperature. This point of view also leads to an interpretation of u β as a “transcendental” Bergman metric.
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| 4. |
- Berman, Robert, 1976
(författare)
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K-polystability of Q-Fano varieties admitting Kahler-Einstein metrics
- 2016
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Ingår i: Inventiones Mathematicae. - : Springer Science and Business Media LLC. - 0020-9910 .- 1432-1297. ; 203:3, s. 973-1025
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Tidskriftsartikel (refereegranskat)abstract
- It is shown that any, possibly singular, Fano variety X admitting a Kahler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of Q-Fano varieties equipped with their anti-canonical polarization. The proof is based on a new formula expressing the Donaldson-Futaki invariants in terms of the slope of the Ding functional along a geodesic ray in the space of all bounded positively curved metrics on the anti-canonical line bundle of X. One consequence is that a toric Fano variety X is K-polystable iff it is K-polystable along toric degenerations iff 0 is the barycenter of the canonical weight polytope P associated to X. The results also extend to the logarithmic setting and in particular to the setting of Kahler-Einsteinmetrics with edge-cone singularities. Applications to geodesic stability, bounds on the Ricci potential and Perelman's lambda-entropy functional on K-unstable Fano manifolds are also given.
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| 5. |
- Berman, Robert, 1976
(författare)
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Large Deviations for Gibbs Measures with Singular Hamiltonians and Emergence of Kahler-Einstein Metrics
- 2017
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Ingår i: Communications in Mathematical Physics. - : Springer Science and Business Media LLC. - 0010-3616 .- 1432-0916. ; 354:3, s. 1133-1172
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Tidskriftsartikel (refereegranskat)abstract
- In the present paper and the companion paper (Berman, Kahler-Einstein metrics, canonical random point processes and birational geometry. arXiv:1307.3634, 2015) a probabilistic (statistical-mechanical) approach to the construction of canonical metrics on complex algebraic varieties X is introduced by sampling "temperature deformed" determinantal point processes. The main new ingredient is a large deviation principle for Gibbs measures with singular Hamiltonians, which is proved in the present paper. As an application we show that the unique Kahler-Einstein metric with negative Ricci curvature on a canonically polarized algebraic manifold X emerges in the many particle limit of the canonical point processes on X. In the companion paper (Berman in 2015) the extension to algebraic varieties X with positive Kodaira dimension is given and a conjectural picture relating negative temperature states to the existence problem for Kahler-Einstein metrics with positive Ricci curvature is developed.
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| 6. |
- Berman, Robert, 1976
(författare)
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On Large Deviations for Gibbs Measures, Mean Energy and Gamma-Convergence
- 2018
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Ingår i: Constructive Approximation. - : Springer Science and Business Media LLC. - 0176-4276 .- 1432-0940. ; 48:1, s. 3-30
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Tidskriftsartikel (refereegranskat)abstract
- We consider the random point processes on a measure space defined by the Gibbs measures associated with a given sequence of N-particle Hamiltonians Inspired by the method of Messer-Spohn for proving concentration properties for the laws of the corresponding empirical measures, we propose a number of hypotheses on that are quite general but still strong enough to extend the approach of Messer-Spohn. The hypotheses are formulated in terms of the asymptotics of the corresponding mean energy functionals. We show that in many situations, the approach even yields a large deviation principle (LDP) for the corresponding laws. Connections to Gamma-convergence of (free) energy type functionals at different levels are also explored. The focus is on differences between positive and negative temperature situations, motivated by applications to complex geometry. The results yield, in particular, large deviation principles at positive as well as negative temperatures for quite general classes of singular mean field models with pair interactions, generalizing the 2D vortex model and Coulomb gases. In a companion paper, the results will be illustrated in the setting of Coulomb and Riesz type gases on a Riemannian manifold X, comparing with the complex geometric setting.
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| 7. |
- Berman, Robert, 1976
(författare)
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On the strict convexity of the K-energy
- 2019
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Ingår i: Pure and Applied Mathematics Quarterly. - 1558-8599 .- 1558-8602. ; 15:4, s. 983-999
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Tidskriftsartikel (refereegranskat)abstract
- Let (X, L) be a polarized projective complex manifold. We show, by a simple toric one-dimensional example, that Mabuchi's K-energy functional on the geodesically complete space of bounded positive (1, 1)-forms in c(1)(L), endowed with the Mabuchi-Donaldson-Semmes metric, is not strictly convex modulo automorphisms. However, under some further assumptions the strict convexity in question does hold in the toric case. This leads to a uniqueness result saying that a finite energy minimizer of the K-energy (which exists on any toric polarized manifold (X, L) which is uniformly K-stable) is uniquely determined modulo automorphisms under the assumption that there exists some minimizer with strictly positive curvature current.
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| 8. |
- Berman, Robert, 1976, et al.
(författare)
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Propagation of chaos, wasserstein gradient flows and toric Kähler-Einstein metrics
- 2018
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Ingår i: Analysis and PDE. - : Mathematical Sciences Publishers. - 2157-5045 .- 1948-206X. ; 11:6, s. 1343-1380
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Tidskriftsartikel (refereegranskat)abstract
- Motivated by a probabilistic approach to Kähler-Einstein metrics we consider a general nonequilibrium statistical mechanics model in Euclidean space consisting of the stochastic gradient flow of a given (possibly singular) quasiconvex N-particle interaction energy. We show that a deterministic "macroscopic" evolution equation emerges in the large N-limit of many particles. This is a strengthening of previous results which required a uniform two-sided bound on the Hessian of the interaction energy. The proof uses the theory of weak gradient flows on the Wasserstein space. Applied to the setting of permanental point processes at "negative temperature", the corresponding limiting evolution equation yields a driftdiffusion equation, coupled to the Monge-Ampère operator, whose static solutions correspond to toric Kähler-Einstein metrics. This drift-diffusion equation is the gradient flow on the Wasserstein space of probability measures of the K-energy functional in Kähler geometry and it can be seen as a fully nonlinear version of various extensively studied dissipative evolution equations and conservation laws, including the Keller-Segel equation and Burger's equation. In a companion paper, applications to singular pair interactions in one dimension are given.
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| 9. |
- Berman, Robert, 1976, et al.
(författare)
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SAMPLING OF REAL MULTIVARIATE POLYNOMIALS AND PLURIPOTENTIAL THEORY
- 2018
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Ingår i: American Journal of Mathematics. - : Project Muse. - 0002-9327 .- 1080-6377. ; 140:3, s. 789-820
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Tidskriftsartikel (refereegranskat)abstract
- We consider the problem of stable sampling of multivariate real polynomials of large degree in a general framework where the polynomials are defined on an affine real algebraic variety M, equipped with a weighted measure. In particular, this framework contains the well-known setting of trigonometric polynomials (when M is a torus equipped with its invariant measure), where the limit of large degree corresponds to a high frequency limit, as well as the classical setting of one-variable orthogonal algebraic polynomials (when M is the real line equipped with a suitable measure), where the sampling nodes can be seen as generalizations of the zeros of the corresponding orthogonal polynomials. It is shown that a necessary condition for sampling, in the general setting, is that the asymptotic density of the sampling points is greater than the density of the corresponding weighted equilibrium measure of M, as defined in pluripotential theory. This result thus generalizes the well-known Landau type results for sampling on the torus, where the corresponding critical density corresponds to the Nyqvist rate, as well as the classical result saying that the zeros of orthogonal polynomials become equidistributed with respect to the logarithmic equilibrium measure, as the degree tends to infinity.
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| 10. |
- Berman, Robert, 1976
(författare)
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Statistical mechanics of interpolation nodes, pluripotential theory and complex geometry
- 2019
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Ingår i: Annales Polonici Mathematici. - : Institute of Mathematics, Polish Academy of Sciences. - 0066-2216 .- 1730-6272. ; 123:1, s. 71-153
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Tidskriftsartikel (refereegranskat)abstract
- This is mainly a survey, explaining how the probabilistic (statistical mechanical) construction of Kahler-Einstein metrics on compact complex manifolds, introduced in a series of works by the author, naturally arises from classical approximation and interpolation problems in C-n. A fair amount of background material is included. Along the way the results are generalized to the non-compact setting of C-n. This yields a probabilistic construction of Kahler solutions to Einstein's equations in C-n, with cosmological constant -beta, from a gas of interpolation nodes in equilibrium at positive inverse temperature beta. In the infinite temperature limit, beta -> 0, solutions to the Calabi-Yau equation are obtained. In the opposite zero temperature case the results may be interpreted as "transcendental" analogs of classical asymptotics for orthogonal polynomials, with the inverse temperature beta playing the role of the degree of a polynomial.
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