Narrow Results

The paper reviews some parts of classical potential theory with applications to two-dimensional fluid dynamics, in particular vortex motion. Energy and forces within a system of point vortices are similar to those for point charges when the vortices are kept fixed, but the dynamics is different in the case of free vortices. Starting from Euler’s and Bernoulli’s equations we derive these laws. Letting the number of vortices tend to infinity leads in the limit to considerations of equilibrium distributions, capacity, harmonic measure and many other notions in potential theory. In particular various kinds of Green functions have a central role in the paper, and we make a distinction between electrostatic and hydrodynamic Green functions. We also consider the corresponding concepts in the case of closed Riemann surfaces provided with a metric. From a canonically defined monopole Green function we rederive much of the classical theory of harmonic and analytic forms on Riemann surfaces. In the final section of the paper we return to the planar case, which then reappears in the form of a symmetric Riemann surface, the Schottky double. Bergman kernels, electrostatic and hydrodynamic, come up naturally as well as other kernels, and associated to the Green function there is a certain Robin function which is important for vortex motion, and which also relates to capacity functions in classical potential theory.

We establish the existence of the asymptotic expansion of the Bergman kernel associated to the spin^c Dirac operators acting on high tensor powers of line bundles with non-degenerate mixed curvature (negative and positive eigenvalues) by extending [15]. We compute the second coefficient b₁ in the asymptotic expansion using the method of [24].

For domains in ℝⁿ we construct the Bergman kernel on the diagonal using solutions of the Dirichlet problem. Starting from this, in a natural way we obtain an algebra 𝔸_n of dimension n(n - 1)/2 + 1 over ℝ and a class of holomorphic functions valued in 𝔸_n. Of course 𝔸₂ is the field of complex numbers, and it turns out that 𝔸₃ is the algebra of quaternions, whereas for n ≥ 4, 𝔸_n is non-associative. Holomorphic functions can be written as f + ω, where f is a (real-valued) function and ω a differential 2-form such that d* ω = df and dω = 0. We investigate the main properties of the obtained objects, especially from the analytic point of view.

We discuss positive closed currents and Fubini–Study currents on orbifolds, as well as Bergman kernels of singular Hermitian orbifold line bundles. We prove that the Fubini–Study currents associated to high powers of a semipositive singular line bundle converge weakly to the curvature current on the set where the curvature is strictly positive, generalizing a well-known theorem of Tian. We include applications to the asymptotic distribution of zeros of random holomorphic sections.

In this paper, we obtain some estimates on the $L^2$-boundary norm of the Bergman kernel for pseudoconvex domains admitting a plurisubharmonic defining function.

Let $X$ be a compact hyperbolic Riemann surface equipped with the Poincaré metric. For any integer $k\ge 2$, we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle ${\mathrm{\Omega }}_X^{\otimes k}$, where ${\mathrm{\Omega }}_X$ is the holomorphic cotangent bundle of $X$. Our first main result estimates the corresponding Bergman metric on $X$ in terms of the Poincaré metric. We then consider a certain natural embedding of the symmetric product of $X$ into a Grassmannian parametrizing subspaces of fixed dimension of the space of all global holomorphic sections of ${\mathrm{\Omega }}_X^{\otimes k}$. The Fubini–Study metric on the Grassmannian restricts to a Kähler metric on the symmetric product of $X$. The volume form for this restricted metric on the symmetric product is estimated in terms of the Bergman kernel of ${\mathrm{\Omega }}_X^{\otimes k}$ and the volume form for the orbifold Kähler form on the symmetric product given by the Poincaré metric on $X$.

We obtain a quantitative estimate of Bergman distance when $\mathrm{\Omega }\subset {ℂ}^n$ is a bounded domain with log-hyperconvexity index ${\alpha }_l(\mathrm{\Omega })>\frac{n-1+\sqrt{(n-1)(n+3)}}{2}$, as well as the $A^2{(logA)}^q$-integrability of the Bergman kernel $K_{\mathrm{\Omega }}(\cdot ,w)$ when ${\alpha }_l(\mathrm{\Omega })>0$.

We relate Berezin–Toeplitz quantization of higher rank vector bundles to quantum-classical hybrid systems and quantization in stages of symplectic fibrations. We apply this picture to the analysis and geometry of vector bundles, including the spectral gap of the Berezin transform and the convergence rate of Donaldson’s iterations toward balanced metrics on stable vector bundles. We also establish refined estimates in the scalar case to compute the rate of Donaldson’s iterations toward balanced metrics on Kähler manifolds with constant scalar curvature.

Let be a C^k-smoothly (with k≥1) bounded pseudoconvex domain and denote its Bergman kernel function. In this article the question is investigated, whether the function is continuous up to the boundary in the topology of the extended real line . We give two counterexamples: one in the class of finite type domains with k = ∞ and one in the class of convex domains with k = 1.

In this lecture results on the Berezin-Toeplitz quantization of arbitrary compact quantizable Kähler manifolds are presented. These results are obtained in joint work with M. Bordemann and E. Meinrenken. The existence of the Berezin-Toeplitz deformation quantization is also covered. Recent results obtained in joint work with A. Karabegov on the asymptotic expansion of the Berezin transform for arbitrary quantizable compact Kähler manifolds are explained. As an application the asymptotic expansion of the Fubini-Study fundamental form under the coherent state embedding is considered. Some comments on the dynamics of the quantum operators are given.

In our previous papers (¹ to ⁶), we studied the Bergman kernels on the "N_p-balls". We represented their Bergman kernels by means of the double series expansion using spherical harmonics. In this note, first we will review our results on the Bergman kernels focused on the two dimensional N_p-balls which are equivalent to the L_p-balls defined by the L_p-norm, and then we will consider an analytic continuation of complex harmonic functions on the N_p-balls through the integral representations with their "harmonic" Bergman kernels.

We explain various results on the asymptotic expansion of the Bergman kernel on Kähler manifolds and also on symplectic manifolds. We also review the "quantization commutes with reduction" phenomenon for a compact Lie group action, and its relation to the Bergman kernel.