Conformal boundary operators, T-curvatures, and conformal fractional Laplacians of odd order

TL;DR

This paper develops new conformally invariant boundary operators of odd order, linking them to fractional Laplacians and curvature quantities, advancing understanding of conformal geometry and representation theory.

Contribution

It introduces parametrized families of boundary operators that generalize known operators and connect to fractional Laplacians and curvature invariants in conformal geometry.

Findings

Constructed conformally invariant boundary operators of odd order.

Linked boundary operators to fractional Laplacians via Dirichlet-to-Neumann maps.

Provided elementary constructions of symmetry breaking intertwinors.

Abstract

We construct continuously parametrised families of conformally invariant boundary operators on densities. These may also be viewed as conformally covariant boundary operators on functions and generalise to higher orders the first-order conformal Robin operator and an analogous third-order operator of Chang-Qing. Our families include operators of critical order on odd-dimensional boundaries. Combined with the (conformal Laplacian power) GJMS operators, a suitable selection of the boundary operators yields formally self-adjoint elliptic conformal boundary problems. Working on a conformal manifold with boundary, we show that the operators yield odd-order conformally invariant fractional Laplacian pseudo-differential operators. To do this, we use higher-order conformally invariant Dirichlet-to-Neumann constructions. We also find and construct new curvature quantities associated to our new operator families. These have links to the Branson Q-curvature and include higher-order generalisations of the mean curvature and the T-curvature of Chang-Qing. In the case of the standard conformal hemisphere, the boundary operator construction is particularly simple; the resulting operators provide an elementary construction of families of symmetry breaking intertwinors between the spherical principal series representations of the conformal group of the equator, as studied by Juhl and others. We use our constructions to shed light on some conjectures of Juhl.