https://neckpinch.com/authors/weiqi-zhang/New arXiv papers by Weiqi Zhang, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2508.13646/arxiv:2508.13646Tue, 19 Aug 2025 00:00:00 GMTWeiqi Zhang, Yashan Zhang — 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 L¹(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 kth eigenvalue of Laplacian operator λₖ≥ c· k^(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.Ricci flowKähler–Ricci flowSingularities(Non)collapsingKähler geometryEigenvalues & spectrum