“…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].…”
Section: The Weighted Ambient Spacementioning
confidence: 99%
“…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…”
Section: Commutation Relations Equation (1) Implies That (2)mentioning
confidence: 99%
“…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.…”
Section: Formal Self-adjointnessmentioning
confidence: 99%
“…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.…”
Section: Formal Self-adjointnessmentioning
confidence: 99%
“…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.…”
We construct weighted GJMS operators on smooth metric measure spaces, and prove that they are formally self-adjoint. We also provide factorization formulas for them in the case of quasi-Einstein spaces and under Gover-Leitner conditions.
“…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].…”
Section: The Weighted Ambient Spacementioning
confidence: 99%
“…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…”
Section: Commutation Relations Equation (1) Implies That (2)mentioning
confidence: 99%
“…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.…”
Section: Formal Self-adjointnessmentioning
confidence: 99%
“…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.…”
Section: Formal Self-adjointnessmentioning
confidence: 99%
“…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.…”
We construct weighted GJMS operators on smooth metric measure spaces, and prove that they are formally self-adjoint. We also provide factorization formulas for them in the case of quasi-Einstein spaces and under Gover-Leitner conditions.
We define weighted renormalized volume coefficients and prove that they are variational. We also prove that they can be written as polynomials of weighted extended obstruction tensors, the weighted Schouten tensor, and the weighted Schouten scalar.
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