Non-linear partial differential equations in conformal geometry

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.