Green's function estimates for compact Kähler manifolds and applications
Weiqi Zhang, Yashan Zhang
TL;DR
The paper establishes nearly optimal uniform integral bounds for Green's functions on compact Kähler manifolds under $L^{1+ε}$ or $L^1(\log L)^{n+ε}$ volume-density conditions, independent of curvature lower bounds. By employing a modified auxiliary complex Monge–Ampère equation, the authors derive both Green's function and gradient bounds and translate them into sharp geometric consequences, including local volume non-collapsing, a Sobolev-type inequality with a logarithmic correction, a diagonal heat-kernel bound, and Weyl-type eigenvalue/eigenfunction estimates with explicit log corrections. These estimates yield improvements for the long-time Kähler-Ricci flow on minimal (and twisted non-minimal) manifolds and extend to general Kähler families, providing new quantitative control that aligns with and extends prior work of Guo-Phong-Song-Sturm, Guedj-Tô, and Vu. The results have significant implications for limit space analysis, diameter control, and spectral geometry in the absence of Ricci curvature lower bounds, and they pave the way for further optimal-exponent investigations and applications to Kähler degenerations. The approach, which foregrounds the critical Green's-function exponent and uses entropy-based functionals, offers a versatile framework for exploring geometric analysis on Kähler manifolds under coarse volume-density hypotheses.
Abstract
Recent works of Guo-Phong-Song-Sturm established for compact Kähler manifolds (even for Kähler spaces of specific singularities) a variety of geometric estimates depending on an upper bound of $L^{1+ε}$ or $L^1(\log L)^{n+ε}$ norms of the volume density but not on any curvature bound, in which a key ingredient is a uniform integral estimate for Green's function. Motivated by their results and further applications, in this paper we shall prove an improved (nearly optimal) integral estimate for Green's function under $L^{1+ε}$ volume density condition, and then apply it to obtain improved global geometric estimates. For instance, one of our results states that the $k$th eigenvalue of Laplacian operator $λ_k\ge c\cdot k^{\frac{1}{n}}(\log k)^{-3}$, where $n$ is the complex dimension of the Kähler manifold and $c$ depends on $n$ and $L^{1+ε}$ norm of the volume density. Also, our results can be applied to the long-time or volume-noncollapsing finite-time Kähler-Ricci flow on compact Kähler manifolds and to a general Kähler family to further extend previous works of Guo-Phong-Song-Sturm, Guedj-Tô and Vu.