Reduced L_{q,p}-Cohomology of Some Twisted Products

Vladimir Golʼdshtein,Yaroslav Kopylov

doi:10.5802/ambp.359

Abstract

Reduced L_{q,p}-Cohomology of Some Twisted Products

Highlights

The Lq,p-cohomology Hqk,p(M ) of a Riemannian manifold (M, g) is defined to be the quotient of the space of closed p-integrable differential k-forms by the exterior differentials of q-integrable k-forms
We prove some vanishing results for the Lq,pcohomology of twisted cylinders [a, b) ×h N for a positive smooth function h : [a, b) × N → R in the case where the base N is a closed manifold and p ≥ q > 1
The reduced Lq,p-cohomology of warped cylinders [a, b) ×h N, i.e., of product manifolds [a, b) × N endowed with a warped product metric g = dt2 + h2(t)gN, where gN is the Riemannian metric of N and h : [a, b) → R is a positive smooth function, was studied by Gol dshtein, Kuz minov, and Shvedov [7], Kuz minov and Shvedov [12, 13], and Kopylov [11] for p, q ∈

Summary

Introduction

The Lq,p-cohomology Hqk,p(M ) of a Riemannian manifold (M, g) is defined to be the quotient of the space of closed p-integrable differential k-forms by the exterior differentials of q-integrable k-forms. We prove some vanishing results for the (reduced) Lq,pcohomology of twisted cylinders [a, b) ×h N for a positive smooth function h : [a, b) × N → R in the case where the base N is a closed manifold and p ≥ q > 1. The reduced Lq,p-cohomology of warped cylinders [a, b) ×h N , i.e., of product manifolds [a, b) × N endowed with a warped product metric g = dt2 + h2(t)gN , where gN is the Riemannian metric of N and h : [a, b) → R is a positive smooth function, was studied by Gol dshtein, Kuz minov, and Shvedov [7], Kuz minov and Shvedov [12, 13] (for p = q), and Kopylov [11] for p, q ∈. The result leads to the vanishing of the “middle-dimensional” cohomology for asymptotic twisted cylinders

Basic Definitions

The Homotopy Operator

The Main Results

Asymptotic Twisted Cylinders