Related papers: Green functions with singularities along complex s…

The Schwarz lemma as one of the most influential results in complex analysis and it has a great impact to the development of several research fields, such as geometric function theory, hyperbolic geometry, complex dynamical systems, and…

Let $\Omega $ be an open, smooth, bounded subset of $ \Bbb R ^n$. In connection with the fractional Laplacian $(-\Delta )^a$ ($a>0$), and more generally for a $2a$-order classical pseudodifferential operator ($\psi $do) $P$ with even…

We prove a Faa di Bruno formula for the Green function in the bialgebra of P-trees, for any polynomial endofunctor P. The formula appears as relative homotopy cardinality of an equivalence of groupoids.

We continue the study of convergence of multipole pluricomplex Green functions for a bounded hyperconvex domain of $\mathbb C^n$, in the case where poles collide. We consider the case where all poles do not converge to the same point in the…

The singularities of the $\Gamma$ function, a meromorphic function on the complex plane, are known to occur at the nonpositive integers. We show, using Euler and Gauss identities, that for all positive integers $n$ and $k$, $$…

Two sharp comparison results are derived for three-dimensional complete noncompact manifolds with scalar curvature bounded from below. The first one concerns the Green's function. When the scalar curvature is nonnegative, it states that the…

Let $u$ be a maximal plurisubharmonic function in a domain $\Omega\subset\mathbb{C}^n$ ($n\geq 2$). It is classical that, for any $U\Subset\Omega$, there exists a sequence of bounded plurisubharmonic functions $PSH(U)\ni u_j\searrow u$…

We solve for the retarded Greens function for linearized gravity in a background with a negative cosmological constant, anti de Sitter space. In this background, it is possible for a signal to reach spatial infinity in a finite time.…

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…

Recent works of Guo-Phong-Song-Sturm established for compact K\"ahler manifolds (even for K\"ahler spaces of specific singularities) a variety of geometric estimates depending on an upper bound of $L^{1+\epsilon}$ or $L^1(\log…

We study the biharmonic equation $\Delta^2 u =u^{-\alpha}$, $0<\alpha<1$, in a smooth and bounded domain $\Omega\subset\RR^n$, $n\geq 2$, subject to Dirichlet boundary conditions. Under some suitable assumptions on $\o$ related to the…

Uniform $L^1$ and lower bounds are obtained for the Green's function on compact K\"ahler manifolds. Unlike in the classic theorem of Cheng-Li for Riemannian manifolds, the lower bounds do not depend directly on the Ricci curvature, but only…

This is a sequel to arXiv:2401.02087. We prove the Green function rigidity conjecture in arXiv:2401.02087 for conformal Laplacian in dimension $n\geq 3$. For the Paneitz operator, we prove the Green function rigidity conjecture when $n\neq…

Let $\phi$ be a psh function on a bounded pseudoconvex open set $\Omega \subset \C^n$, and let ${\cal I}(\phi)$ be the associated multiplier ideal sheaf. Motivated by resolution of singularities issues, we establish an effective version of…

In this paper, we generalize results of Bruinier on automorphic Green functions on Hilbert modular surfaces to arbitrary ideals. For instance, we compute the Fourier expansion of the unregularized Green functions, use it to regularize them,…

We present a calculation of the full retarded Green functions of the Regge-Wheeler and Teukolsky equations obeyed by gravitational field perturbations of Schwarzschild spacetime. We perform the calculations for spacetime points along: (i) a…

The Green function of the spectral ball is constant over the isospectral varieties, is never less than the pullback of its counterpart on the symmetrized polydisk, and is equal to it in the generic case where the pole is a cyclic…

Let $\Gamma$ be a cofinite Fuchsian subgroup. The canonical Green's function associated with $\Gamma$ arises in Arakelov theory when establishing asymptotics for Arakelov invariants of the modular curve associated with some congruence…

We study existence and uniqueness of Green functions for the Cheeger $Q$-Laplacian in metric measure spaces that are Ahlfors $Q$-regular and support a $Q$-Poincar\'e inequality with $Q>1$. We prove uniqueness of Green functions both in the…

In a complete metric space equipped with a doubling measure supporting a $p$-Poincar\'e inequality, we prove sharp growth and integrability results for $p$-harmonic Green functions and their minimal $p$-weak upper gradients. We show that…