Related papers: Zero Lelong number problem

In this article, we report the results we obtained when investigating the numerical solution of some nonlinear eigenvalue problems for the Monge-Amp\`{e}re operator $v\rightarrow \det \mathbf{D}^2 v$. The methodology we employ relies on the…

Let f be a function transcendental and meromorphic in the plane, and define g(z) by g(z) = f(z+1) - f(z). A number of results are proved concerning the existence of zeros of g(z) or g(z)/f(z), in terms of the growth and the poles of f.

We introduce a novel collection of uniqueness problems, with related sampling and interpolation issues. We call them deep zero problems, as they are concerned with local properties at a small number of given points.

The relation between the number of solutions of a nonlinear equation on a Riemannian manifold and the topology of the manifold itself is studied. The technique is based on Ljusternik-Schnirelmann category and Morse theory.

In this paper, we study a geometric property of a continuous plurisubharmonic function which is a solution of the Monge-Amp\`ere equation and has a convex level set. To prove our main theorem, we show a minimum principle of a maximal…

Generalized Lelong numbers of plurisubharmonic functions with respect to plurisubharmonic weights (due to Demailly) are specified for weights with multicircled asymptotics. Explicit formulas for these values are obtained in terms of the…

We consider the "moment vanishing problem" for a general class of piecewise-analytic functions which satisfy on each continuity interval a linear ODE with polynomial coefficients. This problem, which essentially asks how many zero first…

We generalize our recent construction of the zeros of the Riemann $\zeta$-function to two infinite classes of $L$-functions, Dirichlet $L$-functions and those based on level one modular forms. More specifically, we show that there are an…

These notes are concerned with harmonic and holomorphic functions on Euclidean spaces, using quaternions and Clifford algebras in higher dimensions. The main themes are weak solutions, the mean-value property, and subharmonicity.

We give a sufficient condition on the Monge-Amp\`ere mass of a plurisubharmonic function $u$ for $\exp (- 2 u)$ to be locally integrable. This gives a pluripotential theoretic proof of a theorem by J-P. Demailly.

In this article, we characterize plurisubharmonic functions of Lelong number one at the origin, such that the germ of the associated multiplier ideal sheaf is nontrivial: in arbitrary complex dimension, their singularity must be the sum of…

Consider a compact K\"ahler manifold $(X,\omega)$ and the space $\cal E(X,\omega)=\cal E$ of $\omega$--plurisubharmonic functions of full Monge--Amp\`ere mass on it. We introduce a quantity $\rho[u,v]$ to measure the distance between $u,…

We find a relation between the vanishing of a globally defined residue current on $\P^n$ and solution of the membership problem with control of the polynomial degrees. Several classical results appear as special cases, such as Max…

Probability measures with either finite Monge-Amp\`ere energy or finite entropy have played a central role in recent developments in K\"ahler geometry. In this note we make a systematic study of quasi-plurisubharmonic potentials whose…

This is an expository article on the question of whether zero lies in the spectrum of the Laplace-Beltrami operator acting on differential forms on a manifold.

We solve the Dirichlet problem for the complex Monge-Amp\`ere equation on a strictly pseudoconvex with the right hand side being a positive Borel measure which is dominated by the Monge-Amp\`ere measure of a H\"older continuous…

In this note, we solve the complex Monge-Amp\`ere equation for measures with a pluripolar part in compact K\"ahler manifolds. This result generalizes the classical results obtained by Cegrell in bounded hyperconvex domains. We also discuss…

The purpose of this note is to revive in $L^p$ spaces the original A. Markov ideas to study monotonicity of zeros of orthogonal polynomials. This allows us to prove and improve in a simple and unified way our previous result [Electron.…

We study degenerate complex Monge-Amp\`ere equations on a compact K\"ahler manifold $(X,\omega)$. We show that the complex Monge-Amp\`ere operator $(\omega + dd^c \cdot)^n$ is well-defined on the class ${\mathcal E}(X,\omega)$ of…

In this article, we study the nature of zeros of weakly holomorphic modular forms. In particular, we prove results about transcendental zeros of modular forms of higher levels and for certain Fricke groups which extend a work of Kohnen.…