Green's function estimates for compact Kähler manifolds and applications

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 $L^{1+\epsilon }$ or $L^1(\log L{)}^{n+\epsilon }$ 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 $L^{1+\epsilon }$ 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 ${\lambda }_k\ge c\cdot k^{\frac{1}{n}}(\log k{)}^{-3}$, where $n$ is the complex dimension of the Kähler manifold and $c$ depends on $n$ and $L^{1+\epsilon }$ 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.