1. $L^{2}$-Hodge theory on complete almost K\'{a}hler manifold and its application
- Author
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Huang, Teng and Tan, Qiang
- Subjects
Mathematics - Differential Geometry - Abstract
Let $(X,J,\omega)$ be a complete $2n$-dimensional almost K\"{a}hler 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 $\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}$ 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 K\"{a}her form $\omega$. 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 K\"{a}hler manifold with negative sectional curvature., Comment: 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