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$L^{2}$-Hodge theory on complete almost Kähler manifold and its application

Teng Huang, Qiang Tan

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 of the compact almost Kähler manifold with negative sectional curvature.

$L^{2}$-Hodge theory on complete almost Kähler manifold and its application

Abstract

Let be a complete -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 on pure bidegree. In the second part, as some applications of those identities, we establish some vanishing theorems on the spaces of -harmonic -forms on under some growth assumptions on the Käher form . We also give some -estimates to sharpen the vanishing theorems in two specific cases. At last of the article, as an application, we study the topology of the compact almost Kähler manifold with negative sectional curvature.
Paper Structure (15 sections, 31 theorems, 187 equations)

This paper contains 15 sections, 31 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Almost Kähler manifold
  3. $L^{2}$-Hodge theory
  4. de Rham harmonic $(p,q)$-forms
  5. Dolbeault harmonic $(p,q)$-forms
  6. Vanishing theorem for $L^{2}$-harmonic forms
  7. Vanishingly exhaustible $k$-form
  8. Key lemma
  9. Vanishing theorem
  10. The $L^{2}$-estimates
  11. Case one
  12. Case two
  13. Application
  14. $L^{2}$-Betti number
  15. Small Nijenhuis tensor

Key Result

Theorem 1.1

(Generalized Hard Lefschetz Duality) For any complete almost Kähler manifold of dimension $2n$, the operators $\{L,\Lambda,H=[L,\Lambda]\}$ define a finite dimensional representation of $\mathfrak{sl}(2,\mathbb{C})$ on Moreover, for all $0\leq p\leq k\leq n$, are isomorphisms. Furthermore, for any $p,q$ we have an orthogonal direct sum decomposition where

Theorems & Definitions (63)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4: =Theorem \ref{['T5']}+Corollary \ref{['C2']}+Corollary \ref{['C3']}
  • Theorem 1.5
  • Conjecture 1
  • Conjecture 2
  • Theorem 1.6
  • Proposition 2.1
  • proof
  • ...and 53 more