Pluricomplex Green and Lempert functions for equally weighted poles

For $\Omega$ a domain in $\mathbb C^n$, the pluricomplex Green function with poles $a_1, ...,a_N \in \Omega$ is defined as $G(z):=\sup \{u(z): u\in PSH_-(\Omega), u(x)\le \log \|x-a_j\|+C_j \text{when} x \to a_j, j=1,...,N \}$. When there is only one pole, or two poles in the unit ball, it turns out to be equal to the Lempert function defined from analytic disks into $\Omega$ by $L_S (z) :=\inf \{\sum^N_{j=1}\nu_j\log|\zeta_j|: \exists \phi\in \mathcal {O}(\mathbb D,\Omega), \phi(0)=z, \phi(\zeta_j)=a_j, j=1,...,N \}$. It is known that we always have $L_S (z) \ge G_S(z)$. In the more general case where we allow weighted poles, there is a counterexample to equality due to Carlehed and Wiegerinck, with $\Omega$ equal to the bidisk. Here we exhibit a counterexample using only four distinct equally weighted poles in the bidisk. In order to do so, we first define a more general notion of Lempert function "with multiplicities", analogous to the generalized Green functions of Lelong and Rashkovskii, then we show how in some examples this can be realized as a limit of regular Lempert functions when the poles tend to each other. Finally, from an example where $L_S (z) > G_S(z)$ in the case of multiple poles, we deduce that distinct (but close enough) equally weighted poles will provide an example of the same inequality. Open questions are pointed out about the limits of Green and Lempert functions when poles tend to each other.Comment: 25 page

Inverse problem and Bertrand's theorem

The Bertrand's theorem can be formulated as the solution of an inverse problem for a classical unidimensional motion. We show that the solutions of these problems, if restricted to a given class, can be obtained by solving a numerical equation. This permit a particulary compact and elegant proof of Bertrand's theorem.Comment: 11 pages, 3 figure

On Vector Bundles of Finite Order

We study growth of holomorphic vector bundles E over smooth affine manifolds. We define Finsler metrics of finite order on E by estimates on the holomorphic bisectional curvature. These estimates are very similar to the ones used by Griffiths and Cornalba to define Hermitian metrics of finite order. We then generalize the Vanishing Theorem of Griffiths and Cornalba to the Finsler context. We develop a value distribution theory for holomorphic maps from the projectivization of E to projective space. We show that the projectivization of E can be immersed into a projective space of sufficiently large dimension via a map of finite order.Comment: version 2 has some typos corrected; to appear in Manuscripta Mathematic

Powers of ideals and convergence of Green functions with colliding poles

Let us have a family of ideals of holomorphic functions vanishing at N distinct points of a complex manifold, all tending to a single point. As is known, convergence of the ideals does not guarantee the convergence of the pluricomplex Green functions to the Green function of the limit ideal; moreover, the existence of the limit of the Green functions was unclear. Assuming that all the powers of the ideals converge to some ideals, we prove that the Green functions converge, locally uniformly away from the limit pole, to a function which is essentially the upper envelope of the scaled Green functions of the limits of the powers. As examples, we consider ideals generated by hyperplane sections of a holomorphic curve near its singular point. In particular, our result explains recently obtained asymptotics for 3-point models.Comment: 15 pages; typesetting errors fixe

Approximation of conformal mappings using conformally equivalent triangular lattices

Consider discrete conformal maps defined on the basis of two conformally equivalent triangle meshes, that is edge lengths are related by scale factors associated to the vertices. Given a smooth conformal map $f$, we show that it can be approximated by such discrete conformal maps $f^\epsilon$. In particular, let $T$ be an infinite regular triangulation of the plane with congruent triangles and only acute angles (i.e.\ $<\pi/2$). We scale this tiling by $\epsilon>0$ and approximate a compact subset of the domain of $f$ with a portion of it. For $\epsilon$ small enough we prove that there exists a conformally equivalent triangle mesh whose scale factors are given by $\log|f'|$ on the boundary. Furthermore we show that the corresponding discrete conformal maps $f^\epsilon$ converge to $f$ uniformly in $C^1$ with error of order $\epsilon$.Comment: 14 pages, 3 figures; v2 typos corrected, revised introduction, some proofs extende

Available potential vorticity and the wave-vortex decomposition for arbitrary stratification

We consider a rotating non-hydrostatic flow with arbitrary stratification and argue that 1) the appropriate form of potential vorticity (PV) for this system is in terms of isopycnal deviation and 2) the decomposition into energetically orthogonal solutions is fundamentally a PV-inversion. The new closed-form expression for available potential vorticity (APV) is expressed in terms of isopycnal deviation, following the ideas in Wagner & Young (2015). This form of APV linearizes to quasigeostrophic PV (QGPV) after discarding the nonlinear stretching term and a height nonlinearity, the latter of which is not present in constant stratification. This formulation leads to positive definite definitions of potential enstrophy and total energy expressed in terms of isopycnal deviation, from which the quadratic versions emerge at lowest order. It is exactly these quantities diagonalized by the linear eigenmodes. Internal-gravity waves, geostrophic motions, inertial oscillations, and a mean density anomaly form the energetically and enstrophically orthogonal constituents of flow. The complete state of the fluid can be represented in terms of these physically realizeable modes and determined from the derived projection operators using the horizontal velocity and density anomaly. The projection of the fluid state onto the non-hydrostatic wave modes, reveals that one must first account for the PV portion of the flow before recovering the wave solutions. We apply the physical insights of the decomposition to a mesoscale eddy showing how strict adherence to adiabatic rearrangement places strong constraints on the vertical structure of such eddies, including a skew towards stronger cyclonic eddies in the upper water-column. Finally, the expression for APV is shown to reproduce the height nonlinearity of shallow-water PV, a well know feature that breaks the cyclone-anticyclone symmetry in QGPV

Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates

First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ball \B \sub \C^n with its relative logarithmic capacity in \C^n with respect to the same ball \B. An analoguous comparison inequality for Borel subsets of euclidean balls of any generic real subspace of \C^n is also proved. Then we give several interesting applications of these inequalities. First we obtain sharp uniform estimates on the relative size of \psh lemniscates associated to the Lelong class of \psh functions of logarithmic singularities at infinity on \C^n as well as the Cegrell class of \psh functions of bounded Monge-Amp\`ere mass on a hyperconvex domain \W \Sub \C^n. Then we also deduce new results on the global behaviour of both the Lelong class and the Cegrell class of \psh functions.Comment: 25 page

Initial Data for General Relativity with Toroidal Conformal Symmetry

A new class of time-symmetric solutions to the initial value constraints of vacuum General Relativity is introduced. These data are globally regular, asymptotically flat (with possibly several asymptotic ends) and in general have no isometries, but a $U(1)\times U(1)$ group of conformal isometries. After decomposing the Lichnerowicz conformal factor in a double Fourier series on the group orbits, the solutions are given in terms of a countable family of uncoupled ODEs on the orbit space.Comment: REVTEX, 9 pages, ESI Preprint 12