Non-linear partial differential equations in conformal geometry

TL;DR

This paper explores the role of elliptic partial differential equations in conformal geometry, focusing on conformally covariant operators like the family P_{2k} and their invariants, especially in relation to the Yamabe problem.

Contribution

It introduces and analyzes the family of conformally covariant powers of the Laplacian, extending the understanding of PDE methods in conformal geometry.

Findings

Introduction of the P_{2k} operators and their properties

Connection between these operators and conformal invariants

Insights into the Yamabe problem and related PDE techniques

Abstract

In the study of conformal geometry, the method of elliptic partial differential equations is playing an increasingly significant role. Since the solution of the Yamabe problem, a family of conformally covariant operators (for definition, see section 2) generalizing the conformal Laplacian, and their associated conformal invariants have been introduced. The conformally covariant powers of the Laplacian form a family $P_{2k}$ with $k \in \mathbb N$ and $k \leq \frac{n}{2}$ if the dimension $n$ is even. Each $P_{2k}$ has leading order term $(- \Delta)^k$ and is equal to $ (- \Delta) ^k$ if the metric is flat.