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

Franz Berger, Gian Maria Dall'Ara, Duong Ngoc Son
2019 arXiv   pre-print
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
more » ... e 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.
arXiv:1804.07540v2 fatcat:yvveadzkhneijlzirzc6df52ei