Sobolev homeomorphisms and Poincare inequality

Abstract

We study global regularity properties of Sobolev homeomorphisms on $n$-dimensional Riemannian manifolds under the assumption of $p$-integrability of its first weak derivatives in degree $p\ge n-1$. We prove that inverse homeomorphisms have integrable first weak derivatives. For the case $p>n$ we obtain necessary conditions for existence of Sobolev homeomorphisms between manifolds. These necessary conditions based on Poincaré type inequality: $$\underset{c\in \mathbb{R}}{\inf }\Vert u-c\mid L_{\infty }(M)\Vert \le K\Vert u\mid L_{\infty }^1(M)\Vert .$$ As a corollary we obtain the following geometrical necessary condition: {\em If there exists a Sobolev homeomorphisms $\varphi :M\to M^{\prime }$, $\varphi \in W_p^1(M,M^{\prime })$, $p>n$, $J(x,\varphi )\ne 0$ a. e. in $M$, of compact smooth Riemannian manifold $M$ onto Riemannian manifold $M^{\prime }$ then the manifold $M^{\prime }$ has finite geodesic diameter.}}

In the first version, there was an inaccuracy in Theorem 4. In the revised version added additional assumptions