A Liouville theorem for M\"{o}bius invariant equations

TL;DR

This paper classifies second-order M"{o}bius invariant differential operators in 2D and proves a Liouville theorem for related elliptic equations, advancing understanding of conformally invariant PDEs.

Contribution

It provides a complete classification of M"{o}bius invariant second-order operators and establishes a Liouville theorem for their elliptic equations, a novel result in conformal geometry.

Findings

Classification of all second-order M"{o}bius invariant differential operators in 2D

Proof of a Liouville type theorem for these elliptic equations

Enhanced understanding of conformally invariant elliptic PDEs

Abstract

In this paper we classify M\"{o}bius invariant differential operators of second order in two dimensional Euclidean space, and establish a Liouville type theorem for general M\"{o}bius invariant elliptic equations.