| 21. |
- 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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| 22. |
- 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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| 23. |
- Berman, Robert, 1976, et al.
(författare)
-
Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties
- 2013
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Ingår i: Annales de la faculté des sciences de Toulouse. - : Cellule MathDoc/CEDRAM. - 0240-2963 .- 2258-7519. ; 22:4, s. 649-711
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Tidskriftsartikel (refereegranskat)abstract
- We show, using a direct variational approach, that the second boundary value problem for the Monge-Ampère equation in $\mathbb{R}^{n}$ with exponential non-linearity and target a convex body $P$ is solvable iff $0$ is the barycenter of $P.$ Combined with some toric geometry this confirms, in particular, the (generalized) Yau-Tian-Donaldson conjecture for toric log Fano varieties $(X,\Delta )$ saying that $(X,\Delta )$ admits a (singular) Kähler-Einstein metric iff it is K-stable in the algebro-geometric sense. We thus obtain a new proof and extend to the log Fano setting the seminal result of Wang-Zhou concerning the case when $X$ is smooth and $\Delta $ is trivial. Li’s toric formula for the greatest lower bound on the Ricci curvature is also generalized. More generally, we obtain Kähler-Ricci solitons on any log Fano variety and show that they appear as the large time limit of the Kähler-Ricci flow. Furthermore, using duality, we also confirm a conjecture of Donaldson concerning solutions to Abreu’s boundary value problem on the convex body $P$ in the case of a given canonical measure on the boundary of $P.$
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| 24. |
- Berman, Robert, 1976
(författare)
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RELATIVE KAHLER-RICCI FLOWS AND THEIR QUANTIZATION
- 2013
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Ingår i: Analysis & Pde. - : Mathematical Sciences Publishers. - 1948-206X .- 2157-5045. ; 6:1, s. 131-180
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Tidskriftsartikel (refereegranskat)abstract
- Let pi : x -> S be a holomorphic fibration 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 and study their convergence properties. Mainly three different settings are investigated: the case when the fibers are Calabi-Yau manifolds and the case when L = +/- K-X/S is the relative ( anti) canonical line bundle. The main theme studied is whether "positivity in families" is preserved under the flows and its relation to the variation of the moduli of the complex structures of the fibers. 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 pi L-* -> S. Applications to the construction of canonical metrics on the relative canonical bundles of canonically polarized families 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 Kahler-Ricci flow of Donaldson's iteration in a certain double scaling limit.
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| 25. |
- 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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| 26. |
- Berman, Robert, 1976
(författare)
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Sharp Asymptotics for Toeplitz Determinants and Convergence Towards the Gaussian Free Field on Riemann Surfaces
- 2012
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Ingår i: International Mathematics Research Notices. - : Oxford University Press (OUP). - 1073-7928 .- 1687-0247. ; :22, s. 5031-5062
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Tidskriftsartikel (refereegranskat)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. 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 limit, 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 Szego-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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| 27. |
- Berman, Robert, 1976
(författare)
-
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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| 28. |
- Berman, Robert, 1976
(författare)
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Super toeplitz operators on line bundles
- 2006
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Ingår i: Journal of Geometric Analysis. - 1050-6926. ; 16:1, s. 1-22
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Tidskriftsartikel (refereegranskat)abstract
- Let Lk be a high power of a hermitian holomorphic line bundle over a complex manifold X. Given a differential form f on X, we define a super Toeplitz operator Tf acting on the space of harmonic (0, q)-forms with values in Lk, with symbol f. The asymptotic distribution of its eigenvalues, when k tends to infinity, is obtained in terms of the symbol of the operator and the curvature of the line bundle L, given certain conditions on the curvature. For example, already when q = 0, i.e., the case of holomorphic sections, this generalizes a result of Bautet de Monvel and Guillemin to semi-positive line bundles. The asympotics are obtained from the asymptotics of the Bergman kernels of the corresponding harmonic spaces, which have independent interest. Applications to sampling are also given.
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| 29. |
- Berman, Robert, 1976, et al.
(författare)
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Symmetrization of Plurisubharmonic and Convex Functions
- 2014
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Ingår i: Indiana University Mathematics Journal. - : Indiana University Mathematics Journal. - 0022-2518. ; 63:2, s. 345-365
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Tidskriftsartikel (refereegranskat)abstract
- We show that Schwarz symmetrization does not increase the Monge-Ampere energy for S-1-invariant plurisubharmonic functions in the ball. As a result, we derive a sharp Moser-Trudinger inequality for such functions. We also show that similar results do not hold for other balanced domains except for complex ellipsoids, and discuss related questions for convex functions.
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| 30. |
- Berman, Robert, 1976
(författare)
-
The probabilistic vs the quantization approach to Kahler-Einstein geometry
- 2023
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Ingår i: Mathematische Annalen. - 0025-5831 .- 1432-1807.
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Tidskriftsartikel (refereegranskat)abstract
- In the probabilistic construction of Kahler-Einstein metrics on a complex projective algebraic manifold X-involving random point processes on X-a key role is played by the partition function. In this work a new quantitative bound on the partition function is obtained. It yields, in particular, a new direct analytic proof that X admits a Kahler-Einstein metrics if it is uniformly Gibbs stable. The proof makes contact with the quantization approach to Kahler-Einstein geometry.
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