# The Sobolev--Poincar\'e Inequality and the $L_{q,p}$-Cohomology of Twisted Cylinders

## TL;DR

This paper proves a new vanishing theorem for the $L_{q,p}$-cohomology of twisted cylinders, extending previous results for warped cylinders, using advanced Sobolev--Poincaré inequality techniques.

## Contribution

It introduces a vanishing result for $L_{q,p}$-cohomology of twisted cylinders, generalizing known results for warped cylinders with novel methods.

## Key findings

- Vanishing of $L_{q,p}$-cohomology for twisted cylinders.
- Extension of Sobolev--Poincaré inequality methods.
- New results even for warped cylinders.

## Abstract

We establish a vanishing result for the $L_{q,p}$-cohomology ($q\ge p$) of a twisted cylinder, which is a generalization of a warped cylinder. The result is new even for warped cylinders. We base on the methods for proving the $(p,q)$ Sobolev--Poincar\'e inequality developed by L.~Shartser.