Conformal de Rham Hodge theory and operators generalising the Q-curvature

We look at several problems in even dimensional conformal geometry based around the de Rham complex. A leading and motivating problem is to find a conformally invariant replacement for the usual de Rham harmonics. An obviously related problem is to find, for each order of differential form bundle, a ``gauge'' operator which completes the exterior derivative to a system which is both elliptically coercive and conformally invariant. Treating these issues involves constructing a family of new operators which, on the one hand, generalise Branson's celebrated Q-curvature and, on the other hand, compose with the exterior derivative and its formal adjoint to give operators on differential forms which generalise the critical conformal power of the Laplacian of Graham-Jenne-Mason-Sparling. We prove here that, like the critical conformal Laplacians, these conformally invariant operators are not strongly invariant. The construction draws heavily on the ambient metric of Fefferman-Graham and its relationship to the conformal tractor connection and exploring this relationship will be a central theme of the lectures.Comment: 30 pages. Instructional lecture

Scalar Curvature and Projective Compactness

Consider a manifold with boundary, and such that the interior is equipped with a pseudo-Riemannian metric. We prove that, under mild asymptotic non-vanishing conditions on the scalar curvature, if the Levi-Civita connection of the interior does not extend to the boundary (because for example the interior is complete) whereas its projective structure does, then the metric is projectively compact of order 2; this order is a measure of volume growth toward infinity. The result implies a host of results including that the metric satisfies asymptotic Einstein conditions, and induces a canonical conformal structure on the boundary. Underpinning this work is a new interpretation of scalar curvature in terms of projective geometry. This enables us to show that if the projective structure of a metric extends to the boundary then its scalar curvature also naturally and smoothly extends.Comment: Final version to be published in J. Geom. Phys. Includes minor typo corrections and a new summarising corollary. 10 page

Higher symmetries of the conformal powers of the Laplacian on conformally flat manifolds

On locally conformally flat manifolds we describe a construction which maps generalised conformal Killing tensors to differential operators which may act on any conformally weighted tensor bundle; the operators in the range have the property that they are symmetries of any natural conformally invariant differential operator between such bundles. These are used to construct all symmetries of the conformally invariant powers of the Laplacian (often called the GJMS operators) on manifolds of dimension at least 3. In particular this yields all symmetries of the powers of the Laplacian $\Delta^k$, $k\in \mathbb{Z}>0$, on Euclidean space $\mathbb{E}^n$. The algebra formed by the symmetry operators is described explicitly.Comment: 33 pages, minor revisions. To appear in J. Math. Phy

Projective Compactness and Conformal Boundaries

Let $\overline{M}$ be a smooth manifold with boundary $\partial M$ and interior $M$. Consider an affine connection $\nabla$ on $M$ for which the boundary is at infinity. Then $\nabla$ is projectively compact of order $\alpha$ if the projective structure defined by $\nabla$ smoothly extends to all of $\overline{M}$ in a specific way that depends on no particular choice of boundary defining function. Via the Levi--Civita connection, this concept applies to pseudo--Riemannian metrics on $M$. We study the relation between interior geometry and the possibilities for compactification, and then develop the tools that describe the induced geometry on the boundary. We prove that a pseudo--Riemannian metric on $M$ which is projectively compact of order two admits a certain asymptotic form. This form was known to be sufficient for projective compactness, so the result establishes that it provides an equivalent characterization. From a projectively compact connection on $M$, one obtains a projective structure on $\overline{M}$, which induces a conformal class of (possibly degenerate) bundle metrics on the tangent bundle to the hypersurface $\partial M$. Using the asymptotic form, we prove that in the case of metrics, which are projectively compact of order two, this boundary structure is always non--degenerate. We also prove that in this case the metric is necessarily asymptotically Einstein, in a natural sense. Finally, a non--degenerate boundary geometry gives rise to a (conformal) standard tractor bundle endowed with a canonical linear connection, and we explicitly describe these in terms of the projective data of the interior geometry.Comment: Substantially revised, including simpler arguments for many of the main results. 32 pages, comments are welcom