Scattering matrix in conformal geometry

Abstract

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On Fractional GJMS Operators

“…In this section we recall the definition of the fractional GJMS operators via scattering theory [22]. A triple (X n+1 , M n , g + ) is a Poincaré-Einstein manifold if (1) X n+1 is (diffeomorphic to) the interior of a compact manifold X n+1 with boundary ∂X = M n , (2) (X n+1 , g + ) is complete with Ric(g + ) = −ng + , and (3) there exists a nonnegative ρ ∈ C ∞ (X) such that ρ −1 (0) = M n , dρ = 0 along M , and the metric g := ρ 2 g + extends to a smooth metric on X n+1 .…”

“…It is well-known (see [22,30] for more general statements) that given f ∈ C ∞ (M ) and s ∈ C such that Re s > n 2 , s ∈ n 2 + N, and s(n − s) is not in the pure-point spectrum σ pp (−∆ g+ ) of −∆ g+ , the Poisson equation…”

“…[4,6,15]. In this article we are primarily interested in the fractional-order conformally covariant powers of the Laplacian (henceforth fractional GJMS operators) and their associated fractional Q-curvatures as introduced by Graham and Zworski [22]. Unlike familiar geometric objects such as the scalar curvature, the mean curvature, and the conformal Laplacian, it is still not entirely clear what these objects are geometrically.…”

“…The point of working with smooth metric measure spaces is that they provide a natural geometric perspective on the Poisson equations appearing in the work of Graham and Zworski [22]. In Section 4 we show that if (X n+1 , M n , g + ) is a Poincaré-Einstein manifold, then for a given γ ∈ (0, n/2), the Poisson equation which gives rise to the operator P 2γ is precisely the weighted conformal Laplacian of (X n+1 , g + , 1 m0 dvol, m 0 − 1) for m 0 = 1 − 2γ.…”

Conformally invariant powers of the Laplacian — A complete nonexistence theorem

“…Here one seeks to find a metric, from a given conformal class, that has constant scalar curvature. Recently it has become clear that higher order analogues of these operators, viz., conformally invariant operators on weighted functions (i.e., conformal densities) with leading term a power of the Laplacian, have a central role in generating and solving other curvature prescription problems as well as other problems in geometric spectral theory and mathematical physics [2,5,15].…”