The Weighted Ambient Metric

Abstract

We prove the existence and uniqueness of weighted ambient metrics and weighted Poincaré metrics for smooth metric measure spaces.

Weighted GJMS operators on smooth metric measure spaces

“…The existence, and the uniqueness of (g, f ) up to order O(ρ j ) and of [2 −1 g kl (g ρ ) kl + mf −1 (f ρ )] up to order O(ρ j+1 ), has been proven in [7].…”

“…Note that Equation (2) can also be deduced from the fact that Equation ( 1) is a straight weighted ambient metric (cf. [7]). From Equation (2), we get that…”

“…We now prove Theorem 1.2. To begin, we identify the weighted GJMS operators in terms of the weighted Poincaré space [7]. Theorem 6.1.…”

“…Theorem 6.1. Let (X d+1 , g + , f + , m, µ) be a weighted Poincaré space [7] for the smooth metric measure space (M d , g, f, m, µ), and let v ∈ C ∞ (M ). Also, let k ∈ N and s = (d + m)/2 + k, with k ≤ (d + m)/2 if d + m ∈ 2N.…”

“…The ambient metric is a key tool in defining weighted GJMS operators, and a weighted analogue of the ambient metric has recently been defined by Case and the author [7]. By adapting the arguments in [16], we construct the weighted GJMS operators.…”