# Finite distortion Sobolev mappings between manifolds are continuous

## TL;DR

This paper proves that Sobolev mappings of finite distortion between certain Riemannian manifolds are continuous, extending classical results from Euclidean domains to the setting of manifolds, with implications for mappings with positive Jacobian.

## Contribution

It establishes the continuity of finite distortion Sobolev mappings between Riemannian manifolds, generalizing known Euclidean results to a broader geometric context.

## Key findings

- Finite distortion Sobolev mappings are continuous between Riemannian manifolds.
- Mappings with positive Jacobian are continuous.
- Extension of classical Euclidean results to manifold setting.

## Abstract

We prove that if $M$ and $N$ are Riemannian, oriented $n$-dimensional manifolds without boundary and additionally $N$ is compact, then Sobolev mappings $W^{1,n}(M,N)$ of finite distortion are continuous. In particular, $W^{1,n}(M,N)$ mappings with almost everywhere positive Jacobian are continuous. This result has been known since 1976 in the case of mappings $W^{1,n}(\Omega,\mathbb{R}^n)$, where $\Omega\subset\mathbb{R}^n$ is an open set. The case of mappings between manifolds is much more difficult.