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Basic (Dolbeault) Cohomology of Foliated Manifolds with Boundary

Qingchun Ji, Jun Yao

Abstract

In this paper, we develop $L^2$ theory for Riemannian and Hermitian foliations on manifolds with basic boundary. We establish a decomposition theorem, various vanishing theorems, a twisted duality theorem for basic cohomologies and an extension theorem for basic forms of induced Riemannian foliation on the boundary. We prove the complex analogues for Hermitian foliations. To show the Dolbeault decomposition of basic forms, we extend Morrey's basic estimate to foliation version. We also investigate the global regularity for $\bar{\partial}_B$-equations.

Basic (Dolbeault) Cohomology of Foliated Manifolds with Boundary

Abstract

In this paper, we develop theory for Riemannian and Hermitian foliations on manifolds with basic boundary. We establish a decomposition theorem, various vanishing theorems, a twisted duality theorem for basic cohomologies and an extension theorem for basic forms of induced Riemannian foliation on the boundary. We prove the complex analogues for Hermitian foliations. To show the Dolbeault decomposition of basic forms, we extend Morrey's basic estimate to foliation version. We also investigate the global regularity for -equations.
Paper Structure (14 sections, 250 equations)

This paper contains 14 sections, 250 equations.

Table of Contents

  1. Introduction
  2. Riemannian Foliations
  3. Preliminaries of Riemannian foliations
  4. Bochner formula
  5. The basic de Rham-Hodge decomposition
  6. Existence theorems
  7. Duality and extension theorems
  8. Hermitian Foliations
  9. Preliminaries of Hermitian foliations
  10. The basic Dolbeault decomposition
  11. Geometric characterize of basic estimate
  12. Existence theorems
  13. Global regularity of $\bar{\partial}_B$-equations
  14. Extension and duality theorems

Theorems & Definitions (61)

  • Definition 1
  • proof
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  • Remark 1
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  • ...and 51 more