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

Abstract

Abstract. Vanishing results for reduced Lp,q-cohomology are established in the case of twisted products, which are a generalization of warped products. Only the case q ≤ p is considered. This is an extension of some results by Gol ′ dshtein, Kuz ′ minov and Shvedov about the Lp-cohomology of warped cylinders. One of the main observations is the vanishing of the "middledimensional" cohomology for a large class of manifolds.Mathematics Subject Classification. 58A10, 58A12.

The Sobolev–Poincaré inequality and the $L_{q,p}$-cohomology of twisted cylinders

“…If in (1.1) the function h depends only on x then we obtain the familiar notion of a warped product (see [1]). Twisted products were the object of recent investigations [4,6,8,9,10,16,21]. The L q,p -cohomology of warped cylinders [a, b) × h N , i.e., of product manifolds [a, b) × N endowed with a warped product metric…”

“…The paper is organized as follows: In Section 2, we recall some basic definitions concerning the L q,p -cohomology of Riemannian manifolds. Section 3 describes the representations of differential forms on a twisted cylinder obtained in [10] and analogous to the representations of forms on a warped product proposed by Gol ′ dshtein, Kuz ′ minov, and Shvedov in [12]. In Section 4, we develop a version of the weighted Sobolev-Poncaré inequality for convex sets in R n by introducing a homotopy operator and consider some of its consequences; the exposition is based on the ideas of Shartser suggested in [22] and [23].…”