1. $L^{2}$-Hodge theory on complete almost Kähler manifold and its application
- Author
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Huang, Teng and Tan, Qiang
- Subjects
Differential Geometry (math.DG) ,FOS: Mathematics - Abstract
Let $(X,J,ω)$ be a complete $2n$-dimensional almost Kähler manifold. First part of this article, we construct some identities of various Laplacians, generalized Hodge and Serre dualities, a generalized hard Lefschetz duality, and a Lefschetz decomposition, all on the space of $\ker{Δ_{\partial}}\cap\ker{Δ_{\bar{\partial}}}$ on pure bidegree. In the second part, as some applications of those identities, we establish some vanishing theorems on the spaces of $L^{2}$-harmonic $(p,q)$-forms on $X$ under some growth assumptions on the Käher form $ω$. We also give some $L^{2}$-estimates to sharpen the vanishing theorems in two specific cases. At last of the article, as an application, we study the topology and geometry of the compact almost Kähler manifold with negative sectional curvature., 33 pages, Submitted. We have added some references. Since Proposition 4.5 can be obtained directly from Lemma 3.2 in [43], we remove its proof
- Published
- 2023
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