arXiv subjects

Franz Berger

Publications and source records attributed to Franz Berger.

4 recordsLinked to original sources

arXiv2019

Exponential decay of Bergman kernels on complete Hermitian manifolds with Ricci curvature bounded from below

Given a smooth positive measure $μ$ on a complete Hermitian manifold with Ricci curvature bounded from below, we prove a pointwise Agmon-type bound for the corresponding Bergman kernel, under rather general conditions involving the coercivity of an associated complex Laplacian on $(0,1)$-forms. Thanks to an appropriate version of the Bochner--Kodaira--Nakano basic identity, we can give explicit geometric sufficient conditions for such coercivity to hold. Our results extend several known bounds in the literature to the case in which the manifold is neither assumed to be Kähler nor of "bounded geometry". The key ingredients of our proof are a localization formula for the complex Laplacian (of the kind used in the theory of Schrödinger operators) and a mean value inequality for subsolutions of the heat equation on Riemannian manifolds due to Li, Schoen, and Tam. We also show in an appendix that the so-called "twisted basic identities" are standard basic identities with respect to conformally Kähler metrics.

math.CV↗

arXiv2018

Discreteness of spectrum for the $\overline\partial$-Neumann Laplacian on manifolds of bounded geometry

For a Hermitian holomorphic vector bundle over a Hermitian manifold, we consider the Dolbeault Laplacian with $\overline\partial$-Neumann boundary conditions, which is a self-adjoint operator on the space of square-integrable differential forms with values in the given holomorphic bundle. We argue that some known results on the spectral properties of this operator on pseudoconvex domains in $\mathbb C^n$ continue to hold on Kähler manifolds satisfying certain bounded geometry assumptions. In particular, we will consider the Dolbeault complex for forms with values in a line bundle, where known results from magnetic Schrödinger operator theory can be applied.

math.CV↗

arXiv2016

On some spectral properties of the weighted $\overline\partial$-Neumann problem

We derive a necessary condition for compactness of the weighted $\overline\partial$-Neumann operator on the space $L^2(\mathbb C^n,e^{-φ})$, under the assumption that the corresponding weighted Bergman space of entire functions has infinite dimension. Moreover, we compute the essential spectrum of the complex Laplacian for decoupled weights, $φ(z) = φ_1(z_1) + \dotsb + φ_n(z_n)$, and investigate (non-) compactness of the $\overline\partial$-Neumann operator in this case. More can be said if every $Δφ_j$ defines a nontrivial doubling measure.

math.CV↗

arXiv2015

Essential spectra of tensor product Hilbert complexes, and the $\overline\partial$-Neumann problem on product manifolds

We investigate tensor products of Hilbert complexes, in particular the (essential) spectrum of their Laplacians. It is shown that the essential spectrum of the Laplacian associated to the tensor product complex is computable in terms of the spectra of the factors. Applications are given for the $\overline\partial$-Neumann problem on the product of two or more Hermitian manifolds, especially regarding (non-) compactness of the associated $\overline\partial$-Neumann operator.

math.SP↗